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Triangle Congruence
Section 4A
Section 4B
Triangles and Congruence
Proving Triangles Congruent
4-1 Classifying Triangles
4-4 Geometry Lab Explore SSS and SAS Triangle Congruence
4-2 Geometry Lab Develop the Triangle Sum Theorem
4-4 Triangle Congruence: SSS and SAS
4-2 Angle Relationships in Triangles
4-5 Technology Lab Predict Other Triangle Congruence
Relationships
4-3 Congruent Triangles
4-5 Triangle Congruence: ASA, AAS, and HL
4-6 Triangle Congruence: CPCTC
Connecting Geometry to Algebra Quadratic Equations
4-7 Introduction to Coordinate Proof
4-8 Isosceles and Equilateral Triangles
EXTENSION
Proving Constructions Valid
Pacing Guide for 45-Minute Classes
Chapter 4
Countdown to Testing Weeks 7 , 8 , 9
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
4-1 Lesson
4-2 Geometry Lab
4-2 Lesson
4-3 Lesson
Multi-Step Test Prep
Ready to Go On?
4-4 Geometry Lab
DAY 6
DAY 7
4-4 Lesson
4-4 Lesson
DAY 8
DAY 9
4-5 Technology Lab
4-5 Lesson
DAY 10
4-5 Lesson
DAY 11
DAY 12
DAY 13
DAY 14
DAY 15
4-6 Lesson
Connecting Geometry
to Algebra
4-7 Lesson
4-7 Lesson
4-8 Lesson
Multi-Step Test Prep
Ready to Go On?
DAY 16
DAY 17
Chapter 4 Review
EXTENSION
DAY 18
Chapter 4 Test
Pacing Guide for 90-Minute Classes
Chapter 4
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
4-1 Lesson
4-2 Geometry Lab
4-2 Lesson
4-3 Lesson
Multi-Step Test Prep
Ready to Go On?
4-4 Geometry Lab
4-4 Lesson
4-4 Lesson
4-5 Technology Lab
4-5 Lesson
DAY 6
DAY 7
DAY 8
DAY 9
4-6 Lesson
Connecting Geometry
to Algebra
4-7 Lesson
4-7 Lesson
4-8 Lesson
Multi-Step Test Prep
Ready to Go On?
Chapter 4 Review
Chapter 4 Test
212A
Chapter 4
EXTENSION
E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS
DIAGNOSE
Assess
Prior
Knowledge
PRESCRIBE
Before Chapter 4
Diagnose readiness for the chapter.
Prescribe intervention.
Are You Ready? SE p. 213
Are You Ready? Intervention Skills 24, 58, 71
Before Every Lesson
Diagnose readiness for the lesson.
Prescribe intervention.
Warm Up TE, every lesson
Skills Bank SE pp. S50–S81
Reteach CRB, Ch. 1–4
During Every Lesson
Diagnose understanding of lesson concepts.
Prescribe intervention.
Check It Out! SE, every example
Think and Discuss SE, every lesson
Write About It SE, every lesson
Journal TE, every lesson
Questioning Strategies TE, every example
Reading Strategies CRB, every lesson
Success for ELL pp. 41–56
After Every Lesson
Formative
Assessment
Diagnose mastery of lesson concepts.
Prescribe intervention.
Lesson Quiz TE, every lesson
Alternative Assessment TE, every lesson
Test Prep SE, every lesson
Test and Practice Generator
Reteach CRB, every lesson
Problem Solving CRB, every lesson
Test Prep Doctor TE, every lesson
Homework Help Online
Before Chapter 4 Testing
Diagnose mastery of concepts in the chapter.
Prescribe intervention.
Ready to Go On? SE pp. 239, 281
Multi-Step Test Prep SE pp. 238, 280
Section Quizzes AR pp. 65–66
Test and Practice Generator
Ready to Go On? Intervention pp. 41–56
Scaffolding Questions TE pp. 238, 280
Before High Stakes Testing
Diagnose mastery of benchmark concepts.
Prescribe intervention.
College Entrance Exam Practice SE p. 289
Standardized Test Prep SE pp. 292–293
State Test Prep CD-ROM
College Entrance Exam Practice
State Test Prep Workbook
After Chapter 4
Summative
Assessment
Check mastery of chapter concepts.
Prescribe intervention.
Multiple-Choice Tests (Forms A, B, C)
Free-Response Tests (Forms A, B, C)
Performance Assessment AR pp. 67–80
Reteach CRB, every lesson
Lesson Tutorial Videos Chapter 4
Test and Practice Generator
KEY:
Check mastery of benchmark concepts.
Prescribe intervention.
AYP State Tests
College Entrance Exams
State Test Prep Workbook
College Entrance Exam Practice
SE = Student Edition TE = Teacher’s Edition CRB = Chapter Resource Book AR = Assessment Resources
Available online
Available on CD-ROM
212B
RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE
Supporting the Teacher
Chapter 4 Resource Book
Teacher Tools
Practice A, B, C
Power Presentations®
Complete PowerPoint® presentations for Chapter 4 lessons
pp. 3–5, 11–13, 19–21, 27–29, 35–37,
43–45, 51–53, 59–61
Lesson Tutorial Videos®
Reading Strategies %,,
Holt authors Ed Burger and Freddie Renfro present tutorials
to support the Chapter 4 lessons.
pp. 10, 18, 26, 34, 42, 50, 58, 66
One-Stop Planner®
Reteach
pp. 6–7, 14–15, 22–23, 30–31, 38–39, 46–47, 54–55, 62–63
Easy access to all Chapter 4 resources and assessments,
as well as software for lesson planning, test generation,
and puzzle creation
Problem Solving
pp. 9, 17, 25, 33, 41, 49, 57, 65
IDEA Works!®
Challenge
Key Chapter 4 resources and assessments modified to address
special learning needs
pp. 8, 16, 24, 32, 40, 48, 56, 64
Parent Letter pp. 1–2
Lesson Plans........................................................pp. 21–28
Solutions Key ...................................................... Chapter 4
Geometry Posters
Transparencies
TechKeys
Lab Resources
Lesson Transparencies, Volume 2 ......................... Chapter 4
Project Teacher Support
Parent Resources
• Warm Ups
• Teaching Transparencies
• Additional Examples
• Lesson Quizzes
Workbooks
Homework and Practice Workbook
Alternate Openers: Explorations ..........................pp. 21–28
Teacher’s Guide .............................................pp. 21–28
Countdown to Testing ..........................................pp. 13–18
Know-It Notebook
Know-It Notebook ............................................... Chapter 4
Teacher’s Guide ............................................. Chapter 4
• Graphic Organizers
Problem Solving Workbook
Teacher’s Guide .............................................pp. 21–28
State Test Prep Workbook
Teacher’s Guide
Technology Highlights for the Teacher
Power Presentations
One-Stop Planner
Dynamic presentations to engage students.
Complete PowerPoint® presentations for
every lesson in Chapter 4.
KEY:
212C
SE = Student Edition TE = Teacher’s Edition
Chapter 4
%,,
Easy access to Chapter 4 resources and
assessments. Includes lesson-planning, testgeneration, and puzzle-creation software.
English Language Learners
Premier Online Edition
Chapter 4 includes Tutorial Videos,
Lesson Activities, Lesson Quizzes,
Homework Help, and Chapter Project.
JG8E@J? Spanish version available
Available online
Available on CD-ROM
E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS
Reaching All Learners
ENGLISH
LANGUAGE
LEARNERS
Resources for All Learners
English Language Learners
Geometry Lab Activities ....................................... Chapter 4
Are You Ready? Vocabulary.................................. SE p. 213
Technology Lab Activities..................................... Chapter 4
Vocabulary Connections ....................................... SE p. 214
Homework and Practice Workbook .....................pp. 21–28
Lesson Vocabulary .....................................SE, every lesson
Know-It Notebook .............................................. Chapter 4
Vocabulary Exercises .........................SE, every exercise set
Problem Solving Workbook ................................pp. 21–28
Vocabulary Review .............................................. SE p. 284
English Language Learners .................TE pp. 215, 232, 253,
DEVELOPING LEARNERS
Practice A .............................................. CRB, every lesson
Reteach ................................................. CRB, every lesson
Inclusion ....................TE pp. 217, 235, 253, 255, 269, 282
Questioning Strategies ............................ TE, every example
Modified Chapter 4 Resources
................. IDEA Works!
295, 294
Reading Strategies .................................. CRB, every lesson
Success for English Language Learners .................pp. 41–56
Multilingual Glossary
Reaching All Learners Through...
Homework Help Online
Auditory Cues ....................................TE pp. 217, 253, 261
ON-LEVEL LEARNERS
Practice B ............................................... CRB, every lesson
Modeling ...................................................TE pp. 224, 243
Multiple Representations ............................TE pp. 268, 275
Inclusion ....................TE pp. 217, 235, 253, 255, 269, 282
ADVANCED LEARNERS
Practice C ............................................... CRB, every lesson
Challenge ............................................... CRB, every lesson
Reading and Writing Math EXTENSION .......................TE p. 823
Multi-Step Test Prep EXTENSION ......................TE pp. 238, 280
Critical Thinking .................................TE pp. 245, 262, 266
Critical Thinking .................................TE pp. 245, 262, 266
Visual Cues ...............................TE pp. 219, 232, 235, 237,
246–247, 255, 274
Modeling ...................................................TE pp. 224, 243
Multiple Representations ............................TE pp. 268, 275
Test Prep Doctor ................TE pp. 221, 230, 236, 248, 259,
264, 272, 279, 289, 290, 292
Common Error Alerts ..TE pp. 217, 221, 225, 229, 233, 235
243, 257, 263, 269, 271, 277, 279
Scaffolding Questions .................................TE pp. 238, 280
Technology Highlights for Reaching All Learners
Lesson Tutorial Videos
Starring Holt authors Ed Burger and Freddie
Renfro! Live tutorials to support every
lesson in Chapter 4.
KEY:
Multilingual Glossary
Searchable glossary includes definitions
in English, Spanish, Vietnamese, Chinese,
Hmong, Korean, and 4 other languages.
SE = Student Edition TE = Teacher’s Edition CRB = Chapter Resource Book
Online Interactivities
Interactive tutorials provide visually engaging
alternative opportunities to learn concepts and
master skills.
JG8E@J? Spanish version available
Available online
Available on CD-ROM
212D
RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE
Ongoing Assessment
Assessing Prior Knowledge
Daily Assessment
Determine whether students have the prerequisite concepts
and skills for success in Chapter 4.
Are You Ready? JG8E@J?
....................... SE p. 213
Warm Up
................................... TE, every lesson
Provide formative assessment for each day of Chapter 4.
Questioning Strategies ............................ TE, every example
Think and Discuss ......................................SE, every lesson
Check It Out! Exercises ............................SE, every example
Write About It ...........................................SE, every lesson
Journal...................................................... TE, every lesson
Lesson Quiz
............................... TE, every lesson
Alternative Assessment .............................. TE, every lesson
Modified Lesson Quizzes
......................... IDEA Works!
Test Preparation
Provide practice and review for Chapter 4 and standardized
tests.
Multi-Step Test Prep .................................. SE pp. 238, 280
Study Guide: Review .................................. SE pp. 284–287
Test Tackler ............................................... SE pp. 290–291
Standardized Test Prep .............................. SE pp. 292–293
College Entrance Exam Practice ............................ SE p. 289
Countdown to Testing Transparencies
...pp. 13–18
State Test Prep Workbook
State Test Prep CD-ROM
IDEA Works!
Weekly Assessment
Provide formative assessment for each week of Chapter 4.
Multi-Step Test Prep .................................. SE pp. 238, 280
Ready to Go On?
........................ SE pp. 239, 281
Cumulative Assessment ............................. SE pp. 292–293
Test and Practice Generator
.............One-Stop Planner
Formal Assessment
Alternative Assessment
Assess students’ understanding of Chapter 4 concepts
and combined problem-solving skills.
Chapter 4 Project ................................................. SE p. 212
Alternative Assessment .............................. TE, every lesson
Performance Assessment ............................... AR pp. 79–80
Portfolio Assessment ......................................... AR p. xxxiv
Provide summative assessment of Chapter 4 mastery.
Section Quizzes ............................................. AR pp. 65–66
Chapter 4 Test ..................................................... SE p. 288
Chapter Test (Levels A, B, C) ........................... AR pp. 67–78
• Multiple Choice
• Free Response
Cumulative Test ............................................. AR pp. 81–84
Test and Practice Generator
Modified Chapter 4 Test
.............One-Stop Planner
.......................... IDEA Works!
Technology Highlights for Ongoing Assessment
Are You Ready?
KEY:
212E
Ready to Go On?
JG8E@J?
Automatically assess readiness and
prescribe intervention for Chapter 4
prerequisite skills.
Automatically assess understanding
of and prescribe intervention for
Sections 4A and 4B.
SE = Student Edition TE = Teacher’s Edition AR = Assessment Resources
Chapter 4
JG8E@J? Spanish version available
Test and Practice Generator
Use Chapter 4 problem banks to create
assessments and worksheets to print out or
deliver online. Includes dynamic problems.
Available online
Available on CD-ROM
E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS
Formal Assessment
Three levels (A, B, C) of multiple-choice and free-response
chapter tests are available in the Assessment Resources.
A Chapter 4 Test
A Chapter 4 Test
C Chapter 4 Test
C Chapter 4 Test
MULTIPLE CHOICE
FREE RESPONSE
MODIFIED FOR IDEA
B Chapter 4 Test
B Chapter 4 Test
Chapter 4 Test
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you to prove the triangles are congruent
by SAS?
A ⬔D ⬵ ⬔C
B ⬔D ⬵ ⬔A
C ⬔F ⬵ ⬔C
3X 15
4
6
75
7
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8. Which value for x proves that
䉭ABC ⬵ 䉭DEF by SSS?
A 7
B 37
5. Given: 䉭TUV ⬵ 䉭TWV. What is the
value of x ?
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B 123
G1_IDEA_CT04.indd 1
4/11/06 1:49:22 PM
Chapter 4 Test
(continued)
Use the figure for Exercises 10 and 11.
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10. Which postulate or theorem can you use
to prove 䉭ABE ⬵ 䉭CDE ?
A SSS
B SAS
C ASA
11. What additional information will prove
䉭ABE ⬵ _
䉭CDE by either AAS or HL?
_
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_
_
B AE ⬵ CE
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7. Given: ⬔A ⬵ ⬔D, ⬔B ⬵ ⬔E,
⬔C
⬔F, _ _
_
_ ⬵_
_
AB ⬵ DE, BC ⬵ EF, and CA ⬵ FD.
Which is a correct congruence
statement?
A 䉭BCA ⬵ 䉭DEF
B 䉭ABC ⬵ 䉭DEF
4. What is m⬔ACD ?
X
M
B 45
!
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12. To write a coordinate proof, you place a
right isosceles triangle in the coordinate
plane. The legs measure two units. What
is the best place for the vertex angle?
A (0,0)
B (0,2)
C (2, 0)
13. Given: ABCD is a square with vertices
A(0,0), B(0, 4), C (4, 4), and D(4,0). In a
coordinate proof, what
would
_
_ information
be used to prove AB ⬵ CD if you do
NOT use the distance formula?
A x-coordinate of A, x-coordinate of C
B y-coordinate of A, y-coordinate of C
C y-coordinate of A, x-coordinate of C
(2x)°
A 22.5⬚
B 30⬚
C 45⬚
Use the figure for Exercises 15 and 16.
(
*
&
'
15. What
or theorem proves
_ postulate
_
HG ⬵ FG ?
A Isosceles Triangle Theorem
B Converse of Isosceles Triangle
Theorem
16. If 䉭FGJ ⬵ 䉭HGJ, what reason justifies
the statement ⬔HGJ ⬵ ⬔FGJ?
A ASA
B Symmetric. Prop. of ⬵
C CPCTC
#0#4#
G1_IDEA_CT04.indd 2
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Create and customize Chapter 4 Tests. Instantly
generate multiple test versions, answer keys, and
practice versions of test items.
212F
Triangle
Congruence
SECTION
4A Triangles and Congruence
4A
Triangles and
Congruence
On page 238,
students use the
ancient art of paper
folding to make an
origami swan. They answer questions about the sides, angles, and triangles created when a square piece
of paper is folded.
Exercises designed to prepare students for success on the
Multi-Step Test Prep can
be found on pages 220,
229, and 236.
SECTION
4-1
Classifying Triangles
Lab
Develop the Triangle Sum
Theorem
4-2 Angle Relationships in Triangles
4-3 Congruent Triangles
4B Proving Triangle
Congruence
Lab Explore SSS and SAS Triangle
Congruence
4-4 Triangle Congruence: SSS
and SAS
Lab
Predict Other Triangle
Congruence Relationships
4-5
Triangle Congruence: ASA,
AAS, and HL
4B
Proving Triangle
Congruence
4-6 Triangle Congruence: CPCTC
4-7 Introduction to Coordinate Proof
On page 280,
students see how
geometric concepts
are used to design
and construct a doghouse. Students
prove several facts about the triangles used in the design, including
that they are congruent.
Exercises designed to prepare
students for success on
the Multi-Step Test Prep
can be found on pages
247, 258, 264, 271, and
278.
4-8
Isosceles and Equilateral Triangles
Ext
Proving Constructions Valid
When you turn a kaleidoscope,
the shapes flip to form a variety of
designs. You can create flexagons
that also flip to form patterns.
KEYWORD: MG7 ChProj
212
Chapter 4
Flexible Creations
About the Project
Project Resources
In the Chapter Project, students fold strips of
paper to form rows of congruent equilateral
triangles. Then they use these strips to create
their own polygonal flexagons.
All project resources for teachers and
students are provided online.
ge07se_c04_0212_0215.indd 212
Materials:
• paper, scissors, glue, ruler and protractor or straightedge and compass
• geometry software
KEYWORD: MG7 ProjectTS
212
Chapter 4
8/15/05 12:50:12 PM
Vocabulary
Match each term on the left with a definition on the right.
A. a statement that is accepted as true without proof
1. acute angle F
Organizer
2. congruent segments D B. an angle that measures greater than 90° and less than 180°
3. obtuse angle B
C. a statement that you can prove
Objective: Assess students’
4. postulate A
D. segments that have the same length
understanding of prerequisite skills.
5. triangle E
E. a three-sided polygon
F. an angle that measures greater than 0° and less than 90°
Prerequisite Skills
Measure Angles
Measure Angles
Use a protractor to measure each angle.
6.
Solve Equations with Fractions
7.
Connect Words and Algebra
Assessing Prior
Knowledge
35°
90°
INTERVENTION
Diagnose and Prescribe
Use a protractor to draw an angle with each of the following measures.
8. 20°
9. 63°
10. 105°
11. 158°
Use this page to determine
whether intervention is necessary
or whether enrichment is
appropriate.
For exercises 8–11 check students’ drawings.
Solve Equations with Fractions
Solve.
9 x + 7 = 25 4
12. _
2
12 2 3
1 =_
14. x - _
5
5 5
_2
Resources
2 =_
4
13. 3x - _
3 3 3
21 3 1
15. 2y = 5y - _
2
2
_
_
Are You Ready?
Intervention and
Enrichment Worksheets
Are You Ready? CD-ROM
Connect Words and Algebra
Are You Ready? Online
Write an equation for each statement.
16. Tanya’s age t is three times Martin’s age m. t = 3m
17. Twice the length of a segment x is 9 ft. 2x = 9
18. The sum of 53° and twice an angle measure y is 90°. 53 + 2y = 90
19. The price of a radio r is $25 less than the price of a CD player p. r = p - 25
_1 j = b + 5
2
20. Half the amount of liquid j in a jar is 5 oz more than the amount of liquid b in a bowl.
Triangle Congruence
NO
ge07se_c04_0212_0215.indd 213
INTERVENE
213
YES
Diagnose and Prescribe
ENRICH
8/15/05 12:50:45 PM
ARE YOU READY? Intervention, Chapter 4
Prerequisite Skill
Worksheets
CD-ROM
Measure Angles
Skill 24
Activity 24
Solve Equations with Fractions
Skill 71
Activity 71
Connect Words and Algebra
Skill 58
Activity 58
Online
Diagnose and
Prescribe Online
ARE YOU READY?
Enrichment, Chapter 4
Worksheets
CD-ROM
Online
Are You Ready?
213
CHAPTER
Study Guide:
Preview
4
Organizer
Key
Vocabulary/Vocabulario
GI
Objective: Help students
organize the new concepts they
will learn in Chapter 4.
<D
@<I
Previously, you
• measured and classified angles.
• wrote definitions for triangles
and other polygons.
Online Edition
• used deductive reasoning.
• planned and wrote proofs.
Multilingual Glossary
Resources
You will study
• classifying triangles.
• proving triangles congruent.
• using corresponding parts of
Multilingual Glossary Online
KEYWORD: MG7 Glossary
congruent triangles in proofs.
• positioning figures in the
Answers to
Vocabulary Connections
•
1.
coordinate plane for use in
proofs.
proving theorems about
isosceles and equilateral
triangles.
Possible answer: Yes, I think the
is acute.
2. Possible answer: Exterior means
“outside.” An ext. ∠ of a is
located outside the .
You can use the skills
learned in this chapter
• in Algebra 2 and Precalculus.
• in other classes, such as in
3. Possible answer: An obtuse is
a that contains an obtuse ∠.
4. Possible answer:
•
214
Chapter 4
triángulo acutángulo
congruent polygons
polígonos congruentes
corollary
corolario
equilateral triangle
triángulo equilátero
exterior angle
ángulo externo
interior angle
ángulo interno
isosceles triangle
triángulo isósceles
obtuse triangle
triángulo obtusángulo
right triangle
triángulo rectángulo
scalene triangle
triángulo escaleno
Vocabulary Connections
To become familiar with some of the
vocabulary terms in the chapter, consider
the following. You may refer to the chapter,
the glossary, or a dictionary if you like.
1. The Latin word acutus means “pointed”
or “sharp.” Draw a triangle that looks
pointed or sharp. Do you think this is
an acute triangle ?
2. Consider the everyday meaning of
the word exterior. Where do you think
an exterior angle of a triangle is located?
3. You already know the definition of an
obtuse angle. Use this meaning to make
a conjecture about an obtuse triangle .
4. Scalene comes from a Greek word that
means “uneven.” If the sides of a scalene
triangle are uneven, draw an example of
such a triangle.
Chapter 4
ge07se_c04_0212_0215.indd 214
214
Physics when you solve for
various measures of a triangle
and in Geography when you
identify a location using
coordinates.
outside of school to make
greeting cards or to design
jewelry or whenever you
create sets of objects that
have the same size and shape.
acute triangle
8/15/05 12:51:12 PM
CHAPTER
4
Organizer
Reading Strategy: Read Geometry Symbols
Objective: Help students apply
the strategies to understand and
retain key concepts.
!"
!
RAYÊW
ENDPOINT ITH
AT
POINT
Ȝ89:
ANGLE89<W
ITH
VERTEXAT9
MȜ89:
MEASUREOF
ANGLE89<
û
RIGHTANGLE
PI
!"
LINE
NOT*
CONGRUENT
PERPENDICULAR
PLANE
SEGMENT
Ɂ
ʡ
!"
ȸ0
GI
In Geometry we often use symbols to communicate information.
When studying each lesson, read both the symbols and the words slowly and
carefully. Reading aloud can sometimes help you translate symbols into words.
ST
UV
−− −−−
BC ⊥ GH
p→q
Reading Strategies
ENGLISH
LANGUAGE
LEARNERS
ALUEOFÝ
ABSOLUTEV
Reading Strategy:
Read Geometry Symbols
ȡ
Discuss Students benefit from reading geometric symbols and their
meanings aloud. Have students
listen to other students read statements containing geometric symbols. Then have them practice using
symbols, not words, to write the
statements. They can write these on
butcher paper and post them in the
classroom to use as a visual reference later.
PARALLEL
Translated into Words
Line ST is parallel to line UV.
Segment BC is perpendicular to segment GH.
Extend As students work through
Chapter 4, have them present their
proofs to the class and read the
geometric symbols in them. Discuss
what the symbols mean and how
the meanings would differ if the
symbols were changed or left out.
If p, then q.
The measure of angle QRS is 45 degrees.
∠CDE ∠LMN
Angle CDE is congruent to angle LMN.
Online Edition
Chapter 4 Resource Book
]X]
IFTHEN
m∠QRS = 45°
@<I
Resources
Throughout this course, you will use these symbols and combinations of these
symbols to represent various geometric statements.
Symbol Combinations
<D
Try This
Rewrite each statement using symbols.
1. the absolute value of 2 times pi
2. The measure of angle 2 is 125 degrees.
3. Segment XY is perpendicular to line BC.
4. If not p, then not q.
Answers to Try This
1. 2π
2. m∠2 = 125°
−−

3. XY ⊥ BC
Translate the symbols into words.
5. m∠FGH = m∠VWX
6. ZA
TU
7. ∼p → q
bisects ∠TSU.
8. ST
4. ∼p → ∼q
Triangle Congruence
215
5. The measure of angle FGH
equals the measure of angle
VWX.
6. Line ZA is parallel to line TU.
7. If not p, then q.
ge07se_c04_0212_0215.indd 215
8. Ray ST bisects angle TSU.
8/15/05 12:51:15 PM
Reading and Writing Math
215
SECTION
4A Triangles and Congruence
One-Minute Section Planner
Lesson
Lab Resources
Lesson 4-1 Classifying Triangles
Required
• Classify triangles by their angle measures and side lengths.
• Use triangle classification to find angle measures and side lengths.
✔ SAT-10 □
✔ NAEP □
✔ ACT
✔ SAT
✔ SAT Subject Tests
□
□
□
4-2 Geometry Lab Develop the Triangle Sum Theorem
•
Use patty paper to discover the relationship between the measures
of the interior angles of a triangle.
SAT-10 ✔ NAEP
SAT
SAT Subject Tests
✔ ACT
□
□
□
□
Find the measures of interior and exterior angles of triangles.
Apply theorems about the interior and exterior angles of triangles.
✔ SAT-10 ✔ NAEP ✔ ACT
✔ SAT
✔ SAT Subject Tests
□
□
□
□
triangular objects
Geometry Lab Activities
4-2 Lab Recording Sheet
Required
Technology Lab Activities
4-2 Technology Lab
Optional
patty or similar paper
geometry software
Optional
Use properties of congruent triangles.
Prove triangles congruent by using the definition of congruence.
SAT-10 ✔ NAEP
✔ ACT
✔ SAT
✔ SAT Subject Tests
□
Optional
□
Lesson 4-3 Congruent Triangles
•
•
□
straightedge and compass (MK)
□
Lesson 4-2 Angle Relationships in Triangles
•
•
□
Materials
□
□
patterns with congruent
triangles, color pencils, ruler
(MK), tracing paper
MK = Manipulatives Kit
216A
Chapter 4
Section Overview
Properties and Angle Relationships in Triangles
Lessons 4-1, 4-2
A knowledge of the types of triangles and their properties will be needed
throughout geometry and will be helpful in real-world situations.
Triangle Classification
By Angle Measures
By Side Lengths
Acute Triangle
Equiangular Triangle
Three acute angles
Three congruent
acute angles
Right Triangle
Equilateral Triangle
Isosceles Triangle
Three congruent sides
At least two
congruent sides
Obtuse Triangle
Scalene Triangle
One right angle
One obtuse angle
No congruent sides
Example
näÂ
Acute isosceles
näÂ
ÓäÂ
Right scalene
Theorem
For any ABC,
m∠ A + m∠B + m∠C
= 180°.
{
Corollaries
x
Î
For any ABC and
exterior ∠D adjacent to ∠ A,
m∠D = m∠B + m∠C.
For any right ABC
with right ∠C,
m∠ A + m∠B = 90°.
For any equiangular
ABC, m∠ A = m∠B =
m∠C = 60°.
Congruent Triangles
Lesson 4-3
Properties of congruent triangles are used in mathematical proofs.
−−
−−
−−
−−
−−
−−
∠ A ∠D, AB DE
∠B ∠E, BC EF
∠C ∠F, AC DF
ABC DEF
216B
4-1
Organizer
Pacing: Traditional __12 day
Block
__1 day
4
Objectives: Classify triangles
by their angle measures and side
lengths.
GI
Use triangle classification to find
angle measures and side lengths.
<D
@<I
Classifying
Triangles
4-1
Online Edition
Tutorial Videos
Countdown to
Testing Week 7
Who uses this?
Manufacturers use properties of
triangles to calculate the amount of
material needed to make triangular
objects. (See Example 4.)
Objectives
Classify triangles by
their angle measures and
side lengths.
Use triangle classification
to find angle measures
and side lengths.
A triangle is a steel percussion instrument
in the shape of an equilateral triangle.
Different-sized triangles produce
different musical notes when struck
with a metal rod.
Vocabulary
acute triangle
equiangular triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Recall that a triangle () is a polygon
with three sides. Triangles can be
classified in two ways: by their angle
measures or by their side lengths.
Warm Up
Classify each angle.
2. acute
1.
right
3.
Triangle Classification
{äÂ
£xäÂ
By Angle Measures
Acute Triangle
Equiangular Triangle
Right Triangle
Obtuse Triangle
Three acute
angles
Three congruent
acute angles
One right
angle
One obtuse
angle
obtuse
4. If the perimeter is 47, find x
and the length of each side.
ÓÝÊÊÈ
−−
−− −−
AB, BC, and AC are the sides of ABC.
A, B, and C are the triangle’s vertices.
ÝÊÊÎ
ÎÝÊÊn
1
EXAMPLE
x = 5; 8; 16; 23
Classifying Triangles by Angle Measures
Classify each triangle by its angle measures.
Also available on transparency
ÎäÂ
A EHG
∠EHG is a right angle. So EHG is
a right triangle.
ÎäÂ
ÈäÂ
ÈäÂ
B EFH
∠EFH and ∠HFG form a linear pair, so they are supplementary.
Therefore m∠EFH + m∠HFG = 180°. By substitution,
m∠EFH + 60° = 180°. So m∠EFH = 120°. EFH is an
obtuse triangle by definition.
Q: What do you call a tall kettle on
the stove?
A: Hypotenuse!
1. Use the diagram to classify FHG by its angle measures.
equiangular
216
Chapter 4 Triangle Congruence
1 Introduce
ge07se_c04_0216_0221.indd 216
E X P L O R AT I O N
Motivate
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YOUWILLSORTTRIANGLESACCORDINGTOTHEIRANGLEMEASURESAND
SIDELENGTHS
,OOKATTHESETRIANGLES"ASEDONTHEIRAPPEARANCESSORTTHE
TRIANGLESBYLISTINGTHEMINTHEAPPROPRIATECOLUMNS!TRIANGLE
MAYBELISTEDINMORETHANONECOLUMN
"
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!
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%
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(
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KEYWORD: MG7 Resources
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3IDES
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1--Ê
216
Chapter 4
Ask students to identify different types of triangles found in the classroom and/or in the real
world, such as musical triangles, art sponges, or
roof tops. Give them models of different types of
triangles and have them measure the sides and
angles. Explain that they will learn how to classify these triangles according to their side lengths
and their angle measures. Triangle models can be
found in the Manipulatives Kit (MK).
$RAW ATRIANGLEOFYOUROWNANDEXPLAINWHICHCOLUMNSTHE
TRIANGLESHOULDBELISTEDIN
$ESCRIBE ANOTHERCATEGORYTHATYOUCOULDADDTOTHETABLE
Explorations and answers are provided in the
Explorations binder.
12/2/05 6:55:31 PM
Triangle Classification
Equilateral Triangle
Isosceles Triangle
Three congruent sides
EXAMPLE
2
""
By Side Lengths
Scalene Triangle
At least two
congruent sides
Students sometimes think that a
triangle is acute if it has one acute
angle. Remind them that for a triangle to be acute, all three of its angles
must be acute.
No congruent sides
Classifying Triangles by Side Lengths
Additional Examples
Classify each triangle by its side lengths.
Example 1
A ABC
−− −−
From the figure, AB AC. So AC = 15,
and ABC is equilateral.
When you look at a
figure, you cannot
assume segments are
congruent based on
their appearance.
They must be marked
as congruent.
£x
B ABD
Classify each triangle by its
angle measures.
£n
£x
x
näÂ
By the Segment Addition Postulate,
BD = BC + CD = 15 + 5 = 20.
Since no sides are congruent, ABD is scalene.
ÓäÂ
scalene
3
Using Triangle Classification
Find the side lengths of the triangle.
Step 1 Find the value of x.
−− −−
JK KL
JK = KL
(4x - 1.3) = (x + 3.2)
3x = 4.5
x = 1.5
xÝÊÊä°Ó
Given
£ääÂ
2. Use the diagram to classify ACD by its side lengths.
EXAMPLE
Ê,,",
,/
£äÂ
A. BDC
obtuse
B. ABD
acute
Example 2
ÝÊÊΰÓ
Classify each triangle by its side
lengths.
{ÝÊÊ£°Î
Def. of segs.
££
£Ó
Substitute (4x - 13) for JK and (x + 3.2) for KL.
Add 1.3 and subtract x from both sides.
Step 2 Substitute 1.5 into the expressions to find the side lengths.
JK = 4x - 1.3
= 4 (1.5) - 1.3 = 4.7
KL = x + 3.2
= (1.5) + 3.2 = 4.7
JL = 5x - 0.2
= 5 (1.5) - 0.2 = 7.3
3. Find the side lengths of
equilateral FGH.
17; 17; 17
A. EHF
isosceles
B. EHG
scalene
Find the side lengths of the
triangle. 23.3; 23.3; 44.5
{ÝÊÊ£ä°Ç ÓÝÊÊÈ°Î
ÓÞÊÊÎ
Example 3
ÎÞÊÊ{
{
£ä
Divide both sides by 3.
xÝÊÊÓ
Also available on transparency
xÞÊÊ£n
4-1 Classifying Triangles
217
2 Teach
12/2/05
Guided Instruction
Inclusion Remind students that
ABC is a triangle and ∠ ABC is
one angle of the triangle.
Questioning Strategies
E X A M P LE
ge07se_c04_0216_0221.indd 217
Explain how to classify a triangle by its
angles as acute, equiangular, right, or
obtuse. Show that triangles can also be
classified by their side lengths as equilateral, isosceles, or scalene. Then explain that
some classifications can occur together.
For example, a triangle can be right and
isosceles.
INTERVENTION
Through Auditory Cues
Have students work with a partner. Ask
one student to describe a triangle classified
by its angles and the other student to draw
a diagram of what it would look like. Then
have them reverse roles and have one
student describe a triangle classified by its
sides and the partner draw a diagram of
what it would look like.
1
• How do you find the measure of
the other angles in the triangles?
6:55:34 PM
E X A M P LES 2 – 3
• What segment length can you
change so that there is an equilateral triangle?
Algebra In Example 3,
stress that students substitute1.5 for x as a way
to find and verify the answer to the
problem.
Lesson 4-1
217
EXAMPLE
4
Music Application
A manufacturer produces
musical triangles by bending
pieces of steel into the shape
of an equilateral triangle.
The triangles are available in
side lengths of 4 inches, 7 inches,
and 10 inches. How many
4-inch triangles can the
manufacturer produce from
a 100 inch piece of steel?
Additional Examples
Example 4
A steel mill produces roof supports by welding pieces of steel
beams into equilateral triangles.
Each side of the triangle is 18
feet long. How many triangles
can be formed from 420 feet of
steel beam? 7
{ʈ˜°
{ʈ˜°
The amount of steel needed to
make one triangle is equal to
the perimeter P of the
equilateral triangle.
Also available on transparency
{ʈ˜°
P = 3 (4)
= 12 in.
To find the number of triangles
that can be made from 100 inches.
of steel, divide 100 by the amount
of steel needed for one triangle.
INTERVENTION
Questioning Strategies
E X AM P LE
4
1 triangles
100 ÷ 12 = 8_
3
• Why do you round the answer
down instead of rounding the
answer up?
There is not enough steel to complete a ninth triangle.
So the manufacturer can make 8 triangles from a
100 in. piece of steel.
Each measure is the side length of an equilateral triangle.
Determine how many triangles can be formed from a
100 in. piece of steel.
4a. 7 in. 4
4b. 10 in. 3
THINK AND DISCUSS
1. For DEF, name the three pairs of consecutive sides and the vertex
formed by each.
2. Sketch an example of an obtuse isosceles triangle, or explain why it is
not possible to do so.
3. Is every acute triangle equiangular? Explain and support your answer
with a sketch.
4. Use the Pythagorean Theorem to explain why you cannot draw an
equilateral right triangle.
5. GET ORGANIZED Copy and complete
the graphic organizer. In each box,
describe each type of triangle.
/Àˆ>˜}i
>ÃÈvˆV>̈œ˜
ÞÊÈ`iÃ
218
Chapter 4 Triangle Congruence
Answers to Think and Discuss
3 Close
Summarize
−− −−
−− −−
−− −−
1. DE, EF, E; EF, FD, F; FD, DE, D
ge07se_c04_0216_0221.indd 218
Review triangle classifications by angle
measures and side lengths. Draw an
example of each. Emphasize what sides
or angles need to be congruent to classify
each triangle.
and INTERVENTION
Diagnose Before the Lesson
4-1 Warm Up, TE p. 216
Monitor During the Lesson
Check It Out! Exercises, SE pp. 216–218
Questioning Strategies, TE pp. 217–218
Assess After the Lesson
4-1 Lesson Quiz, TE p. 221
Alternative Assessment, TE p. 221
218
Chapter 4
ÞÊ>˜}iÃ
2. Possible answer:
3. No; all 3 in an acute must be
acute, but they do not have to have the
same measure; possible answer:
4. In an equil. rt. , all 3 sides have the
same length. By the Pyth. Thm., the 3
sides lengths are related by the formula
c 2 = a 2 + b 2, making the hyp. c greater
than either a or b. So the 3 sides cannot have the same length.
5. See p. A3.
12/2/05 6:55:36 PM
4-1
4-1
Exercises
KEYWORD: MG7 4-1
Exercises
KEYWORD: MG7 Parent
GUIDED PRACTICE
Assignment Guide
Vocabulary Apply the vocabulary from this lesson to answer each question.
SEE EXAMPLE
1
p. 216
1. In JKL, JK, KL, and JL are equal. How does this help you classify JKL by
its side lengths? An equilateral has 3 sides.
Assign Guided Practice exercises
as necessary.
2. XYZ is an obtuse triangle. What can you say about the types of angles in XYZ?
If you finished Examples 1–2
Basic 12–17, 20, 23–31,
35–37
Average 12–17, 20, 23–31,
35–37, 39
Advanced 12–17, 20, 23–31,
35–37, 39, 46
One of the is obtuse, and the other 2 are acute.
Classify each triangle by its angle measures.
ΣÂ
ÇäÂ
3. DBC
4. ABD
5. ADC
rt.
rt.
obtuse
x™Â
ÓäÂ
SEE EXAMPLE
2
Classify each triangle by its side lengths.
6. EGH
p. 217
7. EFH
isosc.
8. HFG
scalene
scalene
3
p. 217
n
Ç°{
SEE EXAMPLE
Î If you finished Examples 1–4
Basic 12–29, 35, 36, 39–44,
49–58
Average 12–23, 24–32 even,
33, 34, 38–44, 48–58
Advanced 12–22, 24–30 even,
32–34, 38–58
Multi-Step Find the side lengths of each triangle.
10.
9.
ÈÞ
{ÞÊÊ£Ó
{ÝÊÊä°x
ÝÊÊÓ°{
ÎÊV“
36; 36; 36
SEE EXAMPLE 4
p. 218
ÓÝÊÊ£°Ç
3.1; 3.1; 3.3
11. Crafts A jeweler creates triangular earrings by bending
pieces of silver wire. Each earring is an isosceles triangle
with the dimensions shown. How many earrings can be
made from a piece of wire that is 50 cm long? 6
Homework Quick Check
Quickly check key concepts.
Exercises: 12, 16, 18, 22, 24, 28
£°xÊV“
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
12–14
15–17
18–20
21–22
1
2
3
4
Extra Practice
Skills Practice p. S10
Application Practice p. S31
Classify each triangle by
its angle measures.
ÈäÂ
12. BEA rt.
13. DBC obtuse
14. ABC equiangular
ÎäÂ
Visual For Exercises
15–17, introduce the orientation of isosceles RSP.
Point out that the triangle is not
“upside down” but that the orientation is just different.
ÈäÂ
ÎäÂ
ÎäÂ
ÈäÂ
For Exercise 20, have students use
a different color for each side of
the triangle so they can refer to the
angles by the colors that form them.
*
Classify each triangle by its side lengths.
15. PST
16. RSP
equil.
isosc.
scalene
Multi-Step Find the side lengths of each triangle.
18.
âÊÊx
8; 8; 8
{âÊÊ{
£Ç
17. RPT
19.
,
£ä
-
/
8.6; 8.6
ÓÝÊÊÈ°n
ÎâÊÊ£
20. Check students’
nÝÊÊ£°{
drawings.
−− −− −−
20. Draw a triangle large enough to measure. Label the vertices X, Y, and Z.
a. XY, YZ, XZ,
a. Name the three sides and three angles of the triangle.
∠X, ∠Y, ∠Z
b. Possible answer:
b. Use a ruler and protractor to classify the triangle by its side lengths
scalene obtuse
and angle measures.
4-1 Classifying Triangles
ge07se_c04_0216_0221.indd 219
219
12/2/05 6:55:38 PM
KEYWORD: MG7 Resources
Lesson 4-1
219
Construction
For help with Exercise 39,
have students first construct
−−
AB. Then have them set their compasses to the width AB. Draw an arc
centered at A and then another arc
centered at B. Label the intersection
C and draw ABC.
Carpentry Use the following information for Exercises 21 and 22.
A manufacturer makes trusses, or triangular supports,
for the roofs of houses. Each truss is the shape of an
−− −−
isosceles triangle in which PQ PR. The length of the
−− __4
base QR is 3 the length of each of the congruent sides.
24. Not possible;
an equiangular.
must contain
only acute .
27. Not possible;
an equiangular
must also
be equil.
21. The perimeter of each truss is 60 ft.
Find each side length. 18 ft; 18 ft; 24 ft
*
+
,
22. How many trusses can the manufacturer make from 150 feet of lumber? 2
Draw an example of each type of triangle or explain why it is not possible.
Exercise 40
involves using
the Pythagorean
Theorem to find the length of the
hypotenuse of a right triangle. This
exercise prepares students for the
Multi-Step Test Prep on page 238.
23. isosceles right
24. equiangular obtuse
25. scalene right
26. equilateral acute
27. scalene equiangular
28. isosceles acute
29. An equilateral triangle has a perimeter of 105 in.
What is the length of each side of the triangle? 35 in.
30. ABC
Answers
26.
Classify each triangle by its angles and sides.
Architecture
isosc. obtuse
23.
31. ACD
Ó{Â
isosc. rt.
Ó{Â
25.
32. An isosceles triangle has a perimeter of 34 cm. The congruent sides measure
(4x - 1) cm. The length of the third side is x cm. What is the value of x? 4
28.
33. Architecture The base of the Flatiron Building is a triangle bordered by three
streets: Broadway, Fifth Avenue, and East Twenty-second Street. The Fifth Avenue side
is 1 ft shorter than twice the East Twenty-second Street side. The East Twenty-second
Street side is 8 ft shorter than half the Broadway side. The Broadway side is 190 ft.
a. Find the two unknown side lengths. 173 ft; 87 ft
b. Classify the triangle by its side lengths. scalene
34. No; yes; not every isosc. is
equil. because only 2 of the 3
sides must be . Every equil.
is isosc. because an equil. has 3 sides, and the def. of an
isosc. requires that at least 2
sides be .
35.
34. Critical Thinking Is every isosceles triangle equilateral? Is every equilateral
triangle isosceles? Explain.
Daniel Burnham
designed and built
the 22-story Flatiron
Building in New York
City in 1902.
Tell whether each statement is sometimes, always, or never true. Support your
answer with a sketch.
Source:
www.greatbuildings.com
35. An acute triangle is a scalene triangle. S
36. A scalene triangle is an obtuse triangle. S
36.
37. An equiangular triangle is an isosceles triangle. A
38. Write About It Write a formula for the side length s of an equilateral triangle,
given the perimeter P. Explain how you derived the formula.
37.
39. Construction Use the method for constructing congruent segments to construct
an equilateral triangle. Check students’ constructions.
38. s =
__P . The perimeter of an
3
40. This problem will prepare you for the Multi-Step Test Prep on page 238.
Marc folded a rectangular sheet of paper, ABCD, in half
£äÊV“
−−
along EF. He folded the resulting square diagonally and
then unfolded the paper to create the creases shown.
2 xÊV“
a. Use the Pythagorean Theorem to find DE and CE. 5 √
b. What is the m∠DEC? 90°
c. Classify DEC by its side lengths and by its angle measures.
equil. is 3 times the length of
any 1 side, or P = 3s. Solve this
formula for s by dividing both
sides by 3.
4-1 PRACTICE A
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41. What is the side length of an equilateral triangle with a perimeter of 36__23 inches?
2 inches
1 inches
36_
12_
3
3
1 inches
2 inches
18_
12_
3
9
In Exercise 45, students may
assume that a must be positive.
Explain that the variable a can
represent any real number.
42. The vertices of RST are R(3, 2), S(-2, 3), and T(-2, 1). Which of these best
describes RST?
Isosceles
Scalene
Equilateral
43. Which of the following is NOT a correct
classification of LMN?
Acute
Equiangular
Isosceles
Right
Right
If students chose C
for Exercise 41, they
may be having diffi2
culty dividing __
by 3. In Exercise 44,
3
students may forget that there are
two congruent sides in an isosceles
1
1
triangle. Remind them to add __
x + __
4
2
twice. Make sure that they realize
the variables will add to 0.
ÈäÂ
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ÈäÂ
Îʈ˜°
ÈäÂ
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−− −−
44. Gridded Response ABC is isosceles, and AB AC. AB = _12_x + __14 , and
5_
_
BC = 2 - x . What is the perimeter of ABC? 3
(
(
)
Ê,,",
,/
)
CHALLENGE AND EXTEND
45. A triangle has vertices with coordinates (0, 0), (a, 0), and (0, a), where a ≠ 0.
Classify the triangle in two different ways. Explain your answer.
45. It is an isosc.
since 2 sides
of the have
length a. It is
also a rt. since
2 sides of the lie on the coord.
axes and form
a rt. ∠.
46. Write a two-column proof.
Given: ABC is equiangular.
EF AC
Prove: EFB is equiangular.
Journal
Have students describe at least three
real-world examples of isosceles and
equilateral triangles and include a
sketch or magazine picture of each.
47. Two sides of an equilateral triangle measure (y + 10) units and (y 2 - 2) units.
If the perimeter of the triangle is 21 units, what is the value of y? -3
48. Multi-Step The average length of the
sides of PQR is 24. How much longer
then the average is the longest side? 8
*
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Name the parent function of each function. (Previous course)
49. y = 5x 2 + 4 y = x 2
50. 2y = 3x + 4 y = x
51. y = 2(x - 8)2 + 6 y = x 2
Have students sketch and identify an
example of each type of triangle in
this lesson. Then ask them to verify
their answers by measuring with a
ruler and a protractor (MK).
Determine if each biconditional is true. If false, give a counterexample. (Lesson 2-4)
52. Two lines are parallel if and only if they do not intersect.
F; skew lines do not intersect and are not parallel.
53. A triangle is equiangular if and only if it has three congruent angles. T
4-1
54. A number is a multiple of 20 if and only if the number ends in a 0.
F; possible answer: 30 has a 0 in the ones place, but 30 is not a multiple of 20.
Determine whether each line is parallel to, is perpendicular to, or
coincides with y = 4x. (Lesson 3-6)
55. y = 4x + 2 1 y = 2x coincides
57. _
2
56. 4y = -x + 8 ⊥
1x ⊥
58. -2y = _
2
4-1 Classifying Triangles
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angles and sides.
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Answers
2
46. 1. ABC is equiangular. (Given)
IN
4
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2. ∠ A ∠B ∠C (Def. of
equiangular )
−− −−
3. EF AC (Given)
4. ∠BEF ∠ A, ∠BFE ∠C, (Corr. Post.)
5. ∠BEF ∠ B, ∠BFE ∠B, (Trans.
Prop. of )
È
£ÓäÂ
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È
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2. NQP
obtuse; scalene
3. MNP
acute; scalene
12/2/05 6:55:43 PM
*
4. Find the side lengths of the
triangle. 29; 29; 23
ÎÝÊÊÓ
{ÝÊÊÇ
6. ∠BEF ∠ BFE ( to the same ∠
are .)
7. EFB is equiangular. (Def. of equiangular )
ÓÝÊÊx
Also available on transparency
"
Lesson 4-1
221
4-2
Organizer
Develop the Triangle
Sum Theorem
Use with Lesson 4-2
Pacing:
1
Traditional __
day
2
1
__
Block 4 day
In this lab, you will use patty paper to discover a relationship between
the measures of the interior angles of a triangle.
Objective: Use patty paper to
discover the relationship between
the measures of the interior angles
of a triangle.
Use with Lesson 4-2
Activity
GI
Materials: patty paper
<D
@<I
1 Draw and label ABC on a sheet of
notebook paper.
Online Edition
Countdown to
Testing Week 7
2 On patty paper draw a line and label
a point P on the line.
Resources
Geometry Lab Activities
4-2 Lab Recording Sheet
Teach
Discuss
Discuss the algebraic language used
to describe the relationship between
the angles of a triangle.
3 Place the patty paper on top of the
triangle you drew. Align the papers
−−
so that AB is on line and P and B
coincide. Trace ∠B. Rotate the triangle
and trace ∠C adjacent to ∠B. Rotate
the triangle again and trace ∠A
adjacent to ∠C. The diagram shows
your final step.
Alternative Approach
Have students use geometry software to draw a triangle, measure the
interior angles, and drag the vertices.
Lead them to the conjecture that the
angle sum is always 180°.
Close
Key Concept
Try This
The sum of the measures of the
angles of any triangle is 180°.
When placed together the 3 form a line.
1. What do you notice about the three angles of the triangle that you traced?
2. Repeat the activity two more times using two different triangles. Do you get the
same results each time? yes
Assessment
Journal Have students draw
different-shaped triangles, measure
the angles, and find the sum. Then
compare their results.
3. Write an equation describing the relationship among the measures of the angles
of ABC. m∠A + m∠B + m∠C = 180°
4. Use inductive reasoning to write a conjecture about the sum of the measures of
the angles of a triangle. The sum of the measures of the of a is 180°.
222
Chapter 4 Triangle Congruence
Teacher to Teacher
A pencil can be used to “swing” through
the angles of a triangle. At the final
move the pencil point will face the
opposite direction—a rotation of 180°.
ge07se_c04_0222.indd 222
KEYWORD: MG7 Resources
222
Chapter 4
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8/16/05 5:25:40 PM
Kathleen Kelly
Fairfield, ME
Angle Relationships
in Triangles
4-2
4-2
Organizer
Pacing: Traditional 1 day
Block
of interior and exterior angles of
triangles.
Apply theorems about the interior
and exterior angles of triangles.
Triangulation is a method
used in surveying. Land is
divided into adjacent triangles.
By measuring the sides and
angles of one triangle and
applying properties of triangles,
surveyors can gather information
about adjacent triangles.
Vocabulary
auxiliary line
corollary
interior
exterior
interior angle
exterior angle
remote interior angle
Theorem 4-2-1
Technology Lab
In Technology Lab Activities
This engraving shows the county
surveyor and commissioners laying
out the town of Baltimore in 1730.
GI
Apply theorems about
the interior and exterior
angles of triangles.
2
Objectives: Find the measures
Who uses this?
Surveyors use triangles
to make measurements
and create boundaries.
(See Example 1.)
Objectives
Find the measures of
interior and exterior
angles of triangles.
__1 day
<D
@<I
Online Edition
Tutorial Videos
Countdown to
Testing Week 7
Triangle Sum Theorem
The sum of the angle measures of a triangle is 180°.
m∠A + m∠B + m∠C = 180°
Warm Up
The proof of the Triangle Sum Theorem uses an auxiliary line. An auxiliary line
is a line that is added to a figure to aid in a proof.
Triangle Sum Theorem
PROOF
Given: ABC
Prove: m∠1 + m∠2 + m∠3 = 180°
{
Proof:
£
Î
2. What is the complement of an
angle with measure 17°? 73°
Ű
Ó x
1. Find the measure of exterior
∠DBA of BCD, if m∠DBC =
30°, m∠C = 70°, and m∠D =
80°. 150°
3. How many lines can be drawn
−−
through N parallel to MP ?
Why? 1; Parallel Post.
À>ÜÊŰÊȡÊ
Ê̅ÀœÕ}…Ê°
Whenever you draw
an auxiliary line,
you must be able to
justify its existence.
Give this as the
reason: Through any
two points there is
exactly one line.
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Q: How many feet are in a yard?
4-2 Angle Relationships in Triangles
223
A: It depends on how many people
are in the yard!
1 Introduce
ge07se_c04_0223_0230.indd 223
E X P L O R AT I O N
Motivate
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12/2/05 6:56:59 PM
Have students trace a circular object to create a
protractor-like shape. Then have them label their
drawings with estimated angle measures every
ten degrees. Have students use the protractor
they created to measure and find the sum of the
interior angles of a triangle. Finally have them
draw an exterior angle of a triangle and measure
it. Students should check their estimates with an
actual protractor.
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Explorations and answers are provided in the
Explorations binder.
KEYWORD: MG7 Resources
Lesson 4-2
223
EXAMPLE
1
Surveying Application
The map of France commonly
used in the 1600s was significantly
revised as a result of a triangulation
land survey. The diagram shows
part of the survey map. Use the
diagram to find the indicated
angle measures.
Additional Examples
Example 1
After an accident, the
positions of cars are measured
by law enforcement to
investigate the collision. Use
the diagram drawn from the
information collected to find
the indicated angle measures.
m∠KMN + m∠MNK + m∠NKM = 180°
88 + 48 + m∠NKM = 180
136 + m∠NKM = 180
m∠NKM = 44°
ÈÓÂ
{äÂ
8
Step 1 Find m∠JKL.
m∠NKM + m∠MKJ + m∠JKL = 180°
44 + 104 + m∠JKL = 180
50°
148 + m∠JKL = 180
m∠JKL = 32°
Also available on transparency
Substitute 88 for m∠KMN
and 48 for m∠MNK.
Simplify.
Subtract 136 from both sides.
Lin. Pair Thm. & ∠ Add. Post.
Substitute 44 for m∠NKM
and 104 for m∠MKJ.
Simplify.
Subtract 148 from both sides.
Step 2 Use substitution and then solve for m∠JLK.
m∠JLK + m∠JKL + m∠KJL = 180°
Sum Thm.
m∠JLK + 32 + 70 = 180
Substitute 32 for m∠JKL and
INTERVENTION
70 for m∠KJL.
Questioning Strategies
E X AM P LE
Sum Thm.
B m∠JLK
<
78°
B. m∠YWZ
48°
A m∠NKM
£ÓÂ
A. m∠XYZ
104°
88°
9
7
70°
m∠JLK + 102 = 180
m∠JLK = 78°
1
Simplify.
Subtract 102 from both sides.
1. Use the diagram to find m∠MJK. 32°
• What properties do you need to
know about triangles when using
the triangulation method to calculate the measures of angles?
A corollary is a theorem whose proof follows directly from another theorem.
Here are two corollaries to the Triangle Sum Theorem.
Language Arts The word
auxiliary means “giving
assistance.” Emphasize that
an auxiliary line drawn from a vertex
of a triangle to the opposite side
may assist with a proof.
Corollaries
COROLLARY
4-2-2
HYPOTHESIS
The acute angles of
a right triangle are
complementary.
4-2-3
The measure of
each angle of
an equiangular
triangle is 60°.
CONCLUSION
∠D and ∠E are
complementary.
m∠D + m∠E = 90°
m∠A = m∠B = m∠C = 60°
You will prove Corollaries 4-2-2 and 4-2-3 in Exercises 24 and 25.
224
Chapter 4 Triangle Congruence
2 Teach
ge07se_c04_0223_0230.indd 224
12/2/05 6:57:02 PM
Guided Instruction
Review the method of constructing a parallel line before proving the Triangle Sum
Theorem. Explain why an auxiliary line was
added to the figure in the proof. Compare
the two corollaries to the Triangle Sum
Theorem. Review the meanings of interior,
exterior, and remote before introducing
the Exterior Angle Theorem. Show how
the Third Angles Theorem is related to the
Triangle Sum Theorem.
Through Modeling
Have students cut a triangle out of a
sheet of paper and tear off all three
corners. Have them place these next to
each other to form a line. Then ask
students to explain how this models
the Triangle Sum Theorem.
>
224
Chapter 4
> L V
L
V
EXAMPLE
2
Finding Angle Measures in Right Triangles
""
Ê,,",
,/
One of the acute angles in a right triangle measures 22.9°. What is the
measure of the other acute angle?
Let the acute angles be ∠M and ∠N, with m∠M = 22.9°.
m∠M + m∠N = 90
Acute of rt. are comp.
22.9 + m∠N = 90
Substitute 22.9 for m∠M.
m∠N = 67.1°
Subtract 22.9 from both sides.
For Check It Out Problem 2b, students may write (x – 90)°. Show
them that if the given angle has a
measure of 10° for example, the
remaining acute angle would have
measure (90 - 10)°, not (10 - 90)°.
The measure of one of the acute angles in a right triangle is
given. What is the measure of the other acute angle?
2 ° 41 3 °
2a. 63.7° 26.3°
2b. x° (90 - x)° 2c. 48_
5
5
_
The interior is the set of all points inside the figure. The exterior is the set
of all points outside the figure. An interior angle is formed by two sides of
a triangle. An exterior angle is formed by one side of the triangle and the
extension of an adjacent side. Each exterior angle has two remote interior angles.
A remote interior angle is an interior angle that is not adjacent to the
exterior angle.
Ó
ÝÌiÀˆœÀ
∠4 is an exterior angle.
Its remote interior
angles are ∠1 and ∠2.
˜ÌiÀˆœÀ
£
Î
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Additional Examples
Example 2
One of the acute angles in a right
triangle measures 2x°. What is
the measure of the other acute
angle? (90 - 2x)°
Example 3
Find m∠B.
55°
­ÓÝÊÊήÂ
£xÂ
Theorem 4-2-4
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal
to the sum of the measures of its remote interior angles.
Also available on transparency
Ó
m∠4 = m∠1 + m∠2
­xÝÊÊÈä®Â
£
Î
{
You will prove Theorem 4-2-4 in Exercise 28.
INTERVENTION
Questioning Strategies
EXAMPLE
3
Applying the Exterior Angle Theorem
Find m∠J.
m∠J + m∠H = m∠FGH
5x + 17 + 6x - 1 = 126
E X A M P LE
­ÈÝÊÊ£®Â
Ext. ∠ Thm.
£ÓÈÂ
Substitute 5x + 17
­xÝÊʣǮÂ
for m∠J, 6x - 1
for m∠H, and 126 for m∠FGH.
Simplify.
11x + 16 = 126
Subtract 16 from both sides.
11x = 110
Divide both sides by 11.
x = 10
m∠J = 5x + 17 = 5 (10) + 17 = 67°
2
• What corollary to the Triangle Sum
Theorem is used to find the acute
angles of a right triangle? Does this
apply to every right triangle?
E X A M P LE
3
• What is the relationship between
the angle formed by extending one
of the sides of a triangle and each
interior angle?
3. Find m∠ACD. 141° ­ÓâÊÊ£®Â
­ÈâÊʙ®Â
4-2 Angle Relationships in Triangles
ge07se_c04_0223_0230.indd 225
Reading Math Remind students
of the meanings of interior, exterior, and remote. An interior angle
is inside the figure, an exterior angle is outside the figure, and a remote interior angle
is interior and away from the exterior
angle. Relate the idea of a remote interior
angle to a television remote
control that sends a signal
ENGLISH
LANGUAGE
across the room and away
LEARNERS
from you.
225
12/2/05 6:57:03 PM
Lesson 4-2
225
Theorem 4-2-5
THEOREM
Additional Examples
­{ÞÊÓ®Â
80°, 80°
HYPOTHESIS
If two angles of one triangle
are congruent to two angles
of another triangle, then
the third pair of angles
are congruent.
Example 4
Find m∠K and m∠J.
Third Angles Theorem
CONCLUSION
,
∠N ∠T
/
­ÈÞÊÓÊ{ä®Â
You will prove Theorem 4-2-5 in Exercise 27.
EXAMPLE
Also available on transparency
Applying the Third Angles Theorem
Find m∠C and m∠F.
∠C ∠F
m∠C = m∠F
y 2 = 3y 2 - 72
INTERVENTION
Questioning Strategies
E X AM P LE
4
4
• Why is it not necessary to solve for
y to find the missing measures of
the angles?
You can use
substitution to verify
that m∠F = 36°.
m∠F = (3·36 - 72)
= 36°.
ÊÊÞÊÓÊ
ÊÂ
Third Thm.
Def. of .
ÊÎÞÊÓÊÊÇÓ
ÊÂ
Substitute y 2 for m∠C
and 3y 2 - 72 for m∠F.
-2y 2 = -72
Subtract 3y 2 from both sides.
y 2 = 36
Divide both sides by -2.
So m∠C = 36°.
Since m∠F = m∠C, m∠F = 36°.
*
4. Find m∠P and m∠T.
32°; 32°
ÊÊÓÝÊÓÊ
ÊÂ
,
/
Ê{ÝÊÓÊÊÎÓ
ÊÂ
-
THINK AND DISCUSS
1. Use the Triangle Sum Theorem to explain why the supplement of one
of the angles of a triangle equals in measure the sum of the other two
angles of the triangle. Support your answer with a sketch.
2. Sketch a triangle and draw all of its exterior angles. How many exterior
angles are there at each vertex of the triangle? How many total exterior
angles does the triangle have?
3. GET ORGANIZED Copy and complete the graphic organizer.
In each box, write each theorem in words and then draw a diagram
to represent it.
/…iœÀi“
7œÀ`Ã
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/Àˆ>˜}iÊ-ՓÊ/…iœÀi“
ÝÌiÀˆœÀʘ}iÊ/…iœÀi“
/…ˆÀ`ʘ}iÃÊ/…iœÀi“
226
Chapter 4 Triangle Congruence
Answers to Think and Discuss
3 Close
Summarize
ge07se_c04_0223_0230.indd 226
Review the Triangle Sum Theorem, the
Exterior Angle Theorem, and the Third
Angles Theorem. Have several large
triangles as models for each theorem.
Highlight the specific math concept
on each triangle with different colors.
Students should then elaborate and
discuss each theorem.
and INTERVENTION
Diagnose Before the Lesson
4-2 Warm Up, TE p. 223
Monitor During the Lesson
Check It Out! Exercises, SE pp. 224–226
Questioning Strategies, TE pp. 224–226
1. Since ∠3 and ∠4 are supp. , m∠3 +
m∠4 = 180° by def. ∠1 + ∠2 + ∠3 =
180° by the Sum Thm. By the Trans.
Prop. of =, m∠3 + m∠4 = m∠1 +
m∠2 + m∠3. Subtract m∠3 from both
sides. Then m∠4 = m∠1 + m∠2.
2. 2; 6
Assess After the Lesson
4-2 Lesson Quiz, TE p. 230
Alternative Assessment, TE p. 230
226
Chapter 4
3. See p. A4.
12/2/05 6:57:05 PM
4-2
Exercises
4-2 Exercises
KEYWORD: MG7 4-2
KEYWORD: MG7 Parent
GUIDED PRACTICE
Assignment Guide
Vocabulary Apply the vocabulary from this lesson to answer each question.
1. To remember the meaning of remote interior angle, think of a television remote
control. What is another way to remember the term remote? Possible answer: Think
“out of the way”
2. An exterior angle is drawn at vertex E of DEF. What are its remote interior angles?
If you finished Examples 1–2
Basic 15–18, 23, 24, 29–31,
35
Average 15–18, 23–26, 29–31,
35
Advanced 15–18, 23–26, 29–31,
35, 46, 48, 49
∠D; ∠F
3. What do you call segments, rays, or lines that are added to a given diagram?
auxiliary lines
SEE EXAMPLE
1
p. 224
Astronomy Use the following
information for Exercises 4 and 5.
An asterism is a group of stars that is
easier to recognize than a constellation.
One popular asterism is the Summer
Triangle, which is composed of the
stars Deneb, Altair, and Vega.
i˜iL
Ê ÎÞÊÊ£Î
cÊ
6i}>
ÊxÞÊÊx
cÊ
If you finished Examples 1–4
Basic 15–24, 29–32, 35,
40–44, 50–57
Average 15–24, 29–35, 38–44,
49–57
Advanced 15–29, 33–57
4. What is the value of y? 17
5. What is the measure of each
angle in the Summer Triangle?
36°; 80°; 64°
Ì>ˆÀ
SEE EXAMPLE
2
p. 225
The measure of one of the acute angles
in a right triangle is given. What is the
measure of the other acute angle?
7. y ° (90 - y)°
6. 20.8° 69.2°
SEE EXAMPLE
3
p. 225
Assign Guided Practice exercises
as necessary.
Ê ÓÞÊÊÓ
Êc
Homework Quick Check
Quickly check key concepts.
Exercises: 15, 16, 20, 22, 24, 29
_
°
2°
8. 24_
65 1
3
3
Find each angle measure.
10. m∠L 41°
9. m∠M 28°
­ÓÞÊÊÓ®Â
­ÎÞÊÊ£®Â
*
ÇÝÂ
{nÂ
­ÈÝÊÊ£®Â
+
11. In ABC, m∠A = 65°, and the measure of an exterior angle at C is 117°.
Find m∠B and the m∠BCA. 52°; 63°
SEE EXAMPLE 4
12. m∠C and m∠F 100°; 100°
p. 226
13. m∠S and m∠U 89°; 89°
,
-
1
ÊÊ{ÝÊ Ê
ÊÂ
Ó
ÊÎÝÊ ÊÊÓx
ÊÂ
Ó
­{ÝÊʙ®Â
­xÝÊÊ££®Â
/
14. For ABC and XYZ, m∠A = m∠X and m∠B = m∠Y.
Find the measures of ∠C and ∠Z if m∠C = 4x + 7 and m∠Z = 3(x + 5). 39°
4-2 Angle Relationships in Triangles
ge07se_c04_0223_0230.indd 227
227
12/2/05 6:57:08 PM
KEYWORD: MG7 Resources
Lesson 4-2
227
Algebra In Exercises 21
and 22, remind students
that they do not have to
find the value of y or x, only y 2 and
x 2.
PRACTICE AND PROBLEM SOLVING
15
16–18
19–20
21–22
Answers
25. Possible answer:
1. ABC is equiangular. (Given)
2. m∠ A = m∠B = m∠C (Def. of
equiangular)
3. m∠ A + m∠B + m∠C = 180°
( Sum Thm.)
4. m∠ A + m∠ A + m∠ A = 180°
m∠B + m∠B + m∠B = 180°
m∠C + m∠C + m∠C = 180°
(Subst. Prop.)
5. 3m∠ A = 180°, 3m∠B = 180°,
3m∠C = 180° (Simplify)
6. m∠ A = 60°, m∠B = 60°, m∠C
= 60° (Div. Prop. of =)
27.
B
15. Navigation A sailor on ship A measures
the angle between ship B and the pier
and finds that it is 39°. A sailor on ship B
measures the angle between ship A
and the pier and finds that it is 57°.
What is the measure of the angle between
ships A and B? 84°
Independent Practice
For
See
Exercises Example
1
2
3
4
Pier
Ship B
Ship A
39º
57º
Extra Practice
Skills Practice p. S10
The measure of one of the acute angles in a right triangle is given.
What is the measure of the other acute angle?
1 ° 13 3 °
16. 76_
17. 2x° (90 - 2x)°
18. 56.8° 33.2°
4
4
Application Practice p. S31
_
Find each angle measure.
20. m∠C 61°
19. m∠XYZ 162°
­xÝÊÊÓ®Â
9
7
<
­ÈÝÊÊx®Â
­£xÝÊÊ£n®Â
8
­££ÝÊÊ£®Â
­nÝÊÊ{®Â
21. m∠N and m∠P 48°; 48°
22. m∠Q and m∠S 128°; 128°
*
E
+
ÊÊÓÝÊÓÊ
ÊÂ
Ê£ÓÞÊÓÊÊ£{{
ÊÂ
ÊÊÎÞÊÓÊ
ÊÂ
/
A
C
D
F
1. ABC, DEF, ∠ A ∠D, ∠B
∠E (Given)
2. m∠ A + m∠B + m∠C = 180°
( Sum Thm.)
3. m∠C = 180° - m∠ A - m∠B
(Subtr. Prop. of =)
4. m∠D + m∠E + m∠F = 180°
( Sum Thm.)
5. m∠F = 180° - m∠D - m∠E
(Subtr. Prop. of =)
6. m∠ A = m∠D, m∠B = m∠E,
(Def. of )
7. m∠F = 180° - m∠ A - m∠B
(Subst.)
8. m∠F = m∠C (Trans. Prop. of
=)
9. ∠F ∠C (Def. of )
,
ÊÎÝÊÓÊÊÈ{
ÊÂ
-
23. Multi-Step The measures of the angles of a triangle are in the ratio 1 : 4 : 7.
What are the measures of the angles? (Hint: Let x, 4x, and 7x represent the
angle measures.) 15°; 60°; 105°
24. Complete the proof of Corollary 4-2-2.
Given: DEF with right ∠F
Prove: ∠D and ∠E are complementary.
Proof:
Statements
1. DEF with rt. ∠F
?
−−−−
3. m∠D + m∠E + m∠F = 180°
2. b.
Reasons
?
−−−−
2. Def. of rt. ∠
1. a.
a. Given
b. m∠F = 90°
c. Sum Thm.
d. Subst.
e. m∠D + m∠E = 90°
f. Def. of comp. ?
−−−−
?
−−−−
5. Subtr. Prop.
3. c.
4. m∠D + m∠E + 90° = 180°
4. d.
5. e. ?
−−−−
6. ∠D and ∠E are comp.
6. f.
?
−−−−
25. Prove Corollary 4-2-3 using two different methods of proof.
Given: ABC is equiangular.
Prove: m∠A = m∠B = m∠C = 60°
26. Multi-Step The measure of one acute angle in a right triangle is 1__14 times the
measure of the other acute angle. What is the measure of the larger acute angle? 50°
4-2 PRACTICE A
27. Write a two-column proof of the Third Angles Theorem.
4-2 PRACTICE C
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" " Ê ,,",
Given: ABC⁷楴栠數瑥楯爠慮杬攠
ACD
Prove: ACD  A  B
Hint: BCA⁡湤 DCA⁦潲洠愠汩湥慲⁰慩⸩ Find each angle measure.
,/
In Exercise 26, students may not
1
remember how to solve 1__
4
90. Remind them that means 1 ,
1
1
so 1 1__
is 2__
. Then be sure
4
4
9
29. UXW 36°
30. UWY 48°
31. WZX 48°
32. XYZ
6
ÇnÂ
1
42°
x{Â
1
or multiply 90
they divide 90 by 2__
4
7
8
9
4
, not __
, to find .
by __
4
9
<
33. Critical Thinking 坨慴⁩猠瑨攠浥慳畲攠潦⁡湹⁥硴敲楯爠慮杬攠潦⁡渠敱畩慮杵污爠
瑲楡湧汥㼠坨慴⁩猠瑨攠獵洠潦⁴桥⁥硴敲潲⁡湧汥敡獵敳
+
/ 120°; 360°
Exercise 40 involves
folding a sheet of
paper into a given
34.⁆楮搠 SRQⰠ杩癥渠瑨慴
P Ё U Q Ё T
慮搠 RST ㌷⸵° 37.5°
shape. This exercise prepares stu*
1
,
dents for the Multi-Step Test Prep
35. Multi-Step ⁡⁲楧桴⁴楡湧汥Ⱐ潮攠慣畴攠慮杬攠浥慳畲攠楳‴⁴業敳⁴桥瑨敲⁡捵瑥 on page 238.
慮杬攠浥慳畲坨慴⁩猠瑨攠浥慳畲攠潦⁴桥⁳浡汬敲⁡湧汥
18°
36. Aviation ⁳瑵摹⁴桥⁦潲捥猠潦楦琠慮搠摲慧Ⱐ
瑨攠楧桴⁢潴桥牳⁢畩汴⁡⁧汩摥Ⱐ慴瑡捨敤⁴睯
䑲慧
灥猠瑯⁩琬⁡湤⁦汥眠楴楫攠愠歩瑥⸠周敹潤敬敤
瑨攠瑷漠睩湤⁦潲捥猠慳⁴桥敧猠潦⁡⁲楧桴⁴楡湧汥
確
䱩晴
a. 坨慴⁰慲琠潦⁡⁲楧桴⁴楡湧汥⁩猠景浥搠
36a. hyp.
副灥
禺
敡捨⁲灥
b. ° ° 90° 180°
窺
b. 獥⁴桥楡湧汥⁓洠周敯敭⁴漠睲楴攠
c. ° ° 90°;
慮⁥煵慴楯渠敬慴楮朠瑨攠慮杬攠浥慳畲敳
and are comp. 楮⁴桥⁲楧桴⁴楡湧汥
d. ° ° 90°
e. 53°; 127°
c. 業灬楦礠瑨攠敱畡瑩潮⁦潭⁰慲
b⸠坨慴⁩猠瑨攠敬慴楯湳桩瀠扥瑷敥渠
x⁡湤y
d. 獥⁴桥⁅硴敲楯爠䅮杬攠周敯敭⁴漠睲楴攠慮⁥硰敳獩潮⁦潲
z⁩渠瑥浳映
x
e.⁉x ″㞰Ⱐ畳攠畲⁲敳畬瑳⁦潭⁰慲瑳
c⁡湤d⁴漠晩湤
y⁡湤z
39. Check students’ 37.
drawings. The
measures of the ext. 38.
will be the sum of
the pairs of remote
int. : 155°, 65°,
and 140°.
Answers
28. 1. 䅂䌠 with ext. Ġ䅃 (Given)
2. m m mĠ䅃 180° ( Sum Thm.)
3. mĠ䅃 mĠ䅃 180°
(Lin. Pair Thm.)
4. mĠ䅃 180° mĠ䅃
(Subtr. Prop. of )
5. mĠ䅃 (mĠ m mĠ䅃 ) mĠ䅃 (Subst.)
6. mĠ䅃 mĠ m (Simplify)
Estimation 慷⁡⁴楡湧汥⁡湤⁴睯⁥硴敲楯爠慮杬敳⁡琠敡捨⁶敲瑥砮⁅獴業慴攠瑨攠
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ԅ
ԅ
ԅ ԅ
Given:†
AB
†
BDⰠ
BD
†
DC
A Ё C
ԅ ԅ
Prove:†
AD
†
CB
The ext. at the same vertex of
a are vert. . Since vert. are
, the 2 ext. have the same
measure.
̃
̃
̃ ̃
38. 1. 䅂 䉄, 䉄 䑃, Ġ ԁ (Given)
2. Ġ䅂 and Ġ䍄 are rt. .
(Def. of lines)
3. mĠ䅂 m 䍄 (Def. of rt.
)
4. Ġ䅂 ԁ 䍄 (Rt. ą
Thm.)
5. Ġ䅄 ԁ 䍂 (Third Thm.)
̃
̃
6. 䅄 䍂 (Conv. of Alt. Int. Thm.)
37.
39. Write About It 䄠瑲楡湧汥⁨慳⁡湧汥敡獵敳映ㄱ㖰Ⱐ㐰뀬⁡湤′㖰⸠
灬慩渠桯眠瑯⁦楮搠瑨攠浥慳畲敳映瑨攠瑲楡湧汥猠數瑥楯爠慮杬敳
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潮⁰慧攠㈳㠮
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ԅ
a. DCE⁩猠愠楧桴⁡湧汥⸠
FC⁢楳散瑳
DCE
ԅ
慮搠BC⁢楳散瑳
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If students have difficulty with Exercise
43, have them draw
−−
and label ABC with AC extended
to D and then use reasonable angle
measures to estimate the answer.
41. What is the value of x?
19
52
42. Find the value of s.
23
28
Journal
ÝÂ
57
71
34
56
Ç£Â
£ÓnÂ
xnÂ
ÈÈÂ
­ÓÃÊÊ£ä®Â
43. ∠A and ∠B are the remote interior angles of ∠BCD in ABC. Which of these
equations must be true?
Have students describe how they
remember which angles are remote
interior angles and which ones are
exterior angles.
m∠A - 180° = m∠B
m∠A = 90° - m∠B
m∠BCD = m∠BCA - m∠A
m∠B = m∠BCD - m∠A
44. Extended Response The measures of the angles in a triangle are in the ratio
2 : 3 : 4. Describe how to use algebra to find the measures of these angles. Then find
the measure of each angle and classify the triangle.
Have small groups of students compare answers to statements that
are always, sometimes, or never
true. For example, the supplement
of one of the angles of a triangle is
equal to the sum of the other two
angles of the triangle. Then have the
students write a statement to justify
their answers. Encourage students
to draw diagrams to support their
answers.
4-2
CHALLENGE AND EXTEND
45. An exterior angle of a triangle measures 117°. Its remote interior angles measure
(2y 2 + 7)° and (61 - y 2)°. Find the value of y. 7 or -7
46. A rt. is
formed. The 2
same-side int. are supp., so the 2
formed by their
bisectors must be
comp. That means
the remaining ∠
of the must
measure 90°.
xnÂ
52. f(x) = (x - 3)2 + 5
Èʈ˜°
54. ACD isosc.
­ÝÊÊ£Ó®Â
55. BCD scalene
56. ABD scalene ACD is equil.
Ç°x
230
­ÝÊÓÊÊxä®Â
" *
­ÓÞÊÊÈ®Â
­ÞÊÊÎÓ®Â
>Þ½ÃʅœÕÃi
Also available on transparency
Chapter 4
Answers
230
44. Let 2x, 3x, and 4x represent the ∠
measures. The sum of the ∠ measures
of a is 180°, so 2x + 3x + 4x =
180°. Solving the eqn. for the value of
x, yields x = 20. Find each measure by
substituting 20 for x in each expression.
2x = 2(20) = 40; 3x = 3(20) = 60; 4x
= 4(20) = 80. Since all of the measure less than 90°, they are acute by
def. Thus the is acute.
47. Since an ext. ∠ is = to a sum of 2
remote int. , it must be greater than
either ∠. Therefore it cannot be to a
remote int. ∠.
50–52. See p. A15.
230
Ó
,
-̜Ài
œ…˜½ÃʅœÕÃi
{
n
Chapter 4 Triangle Congruence
4. The diagram is a map showing John’s house, Kay’s house,
and the grocery store. What ge07se_c04_0223_0230.indd
is the angle the two houses
make with the store? 30°
­nÞÊÊ£ä®Â
+
{ʈ˜°
57. What if…? If CA = 8, What is the effect on the
classification of ACD?
+
­ÎÝÊÓ®Â
*
{ʈ˜°
Classify each triangle by its side lengths. (Lesson 4-1)
3. Find m∠N and m∠P. 75°, 75°
3
49. In ABC, m∠B is 5° less than 1_12_ times m∠A. m∠C is 5° less than 2__12 times m∠A.
What is m∠A in degrees? 38°
51. f(x) = x 2 + 1
−−−
53. Find the length of NQ. Name the theorem
or postulate that justifies your answer.
(Lesson 1-2) 6 in.; Seg. Add. Post.
124°
_
50. f(x) = 3x - 4
48. Probability The measure of each angle in a triangle is a multiple of 30°.
2
What is the probability that the triangle has at least two congruent angles?
Make a table to show the value of each function when x is -2, 0, 1, and 4.
(Previous course)
3
­ÓÝÊʣȮÂ
47. Critical Thinking Explain why an exterior angle of a triangle cannot be congruent
to a remote interior angle.
SPIRAL REVIEW
1. The measure of one of the
acute angles in a right triangle
2
is 56__
°. What is the measure
3
of the other acute angle?
1°
33__
2. Find m∠ ABD.
46. Two parallel lines are intersected by a transversal. What type of triangle is formed
by the intersection of the angle bisectors of two same-side interior angles? Explain.
(Hint: Use geometry software or construct a diagram of the angle bisectors of two
same-side interior angles.)
5/8/06 12:28:30 PM
Congruent
Triangles
4-3
4-3
Organizer
Pacing: Traditional 1 day
Block
Objectives
Use properties of
congruent triangles.
congruent triangles.
Prove triangles congruent by using
the definition of congruence.
GI
Geometric figures are congruent if
they are the same size and shape.
Corresponding angles and corresponding sides are in the
same position in polygons with an equal number of sides.
Two polygons are congruent polygons if and only if their
corresponding angles and sides are congruent. Thus triangles
that are the same size and shape are congruent.
Vocabulary
corresponding angles
corresponding sides
congruent polygons
2
Objectives: Use properties of
Who uses this?
Machinists used triangles
to construct a model of the
International Space Station’s
support structure.
Prove triangles congruent
by using the definition of
congruence.
__1 day
<D
@<I
Online Edition
Tutorial Videos, Interactivity
Countdown to
Testing Week 7
Properties of Congruent Polygons
CORRESPONDING
ANGLES
DIAGRAM
∠A ∠D
∠B ∠E
Two vertices that
are the endpoints
of a side are called
consecutive vertices.
For example, P and
Q are consecutive
vertices.
ABC DEF
*
-
+
,
<
7
9
8
polygon PQRS polygon WXYZ
∠C ∠F
∠P ∠W
∠Q ∠X
∠R ∠Y
∠S ∠ Z
CORRESPONDING
SIDES
1. Name all sides and angles of
−− −− −−
FGH. FG, GH, FH, ∠F, ∠G, ∠H
−− −−
AB DE
−− −−
BC EF
−− −−
AC DF
2. What is true about ∠K and
∠L? Why? ; Third Thm.
−− −−−
PQ WX
−− −−
QR XY
−− −−
RS YZ
−− −−
PS WZ
RST and XYZ represent the triangles
of the space station’s support structure.
If RST XYZ, identify all pairs
of congruent corresponding parts.
Angles: ∠R ∠X, ∠S ∠Y, ∠T ∠Z
−− −− −− −− −− −−
Sides: RS XY, ST YZ, RT XZ
∠L ∠E, ∠M ∠F,
∠N
∠P −−
∠H,
−− ∠G,
−− −−
LM
EF, MN
FG,
−− −−
−− −−
NP GH, LP EH
Also available on transparency
-
Naming Congruent Corresponding Parts
1
3. What does it mean for two
segments to be congruent?
They have the same length.
To name a polygon, write the vertices in consecutive order. For example, you
can name polygon PQRS as QRSP or SRQP, but not as PRQS. In a congruence
statement, the order of the vertices indicates the corresponding parts.
EXAMPLE
Warm Up
,
<
/
8
Q: What quantity is represented by
three congruent dirty trees?
A: 99. dirty tree + dirty tree +
dirty tree
9
1. If polygon LMNP polygon EFGH, identify all pairs of
corresponding congruent parts.
4-3 Congruent Triangles
231
1 Introduce
ge07se_c04_0231_0237.indd 231
E X P L O R AT I O N
Motivate
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12/2/05 7:11:11 PM
Show students a quilt or other pattern and ask
them to identify all the triangles that look identical in size and shape. Explain that they will learn
the properties used to prove certain triangles are
identical, or congruent. Encourage students to
think of examples where congruent triangles may
be used or seen in the real world, such as in
furniture, buildings, art, and floor tiles.
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$ESCRIBE HOW MANY PAIRS OF MATCHING ANGLES AND MATCHING
Explorations and answers are provided in the
Explorations binder.
KEYWORD: MG7 Resources
Lesson 4-3
231
EXAMPLE
2
Using Corresponding Parts of Congruent Triangles
Given: EFH GFH
Additional Examples
Given: PQR STW. Identify
all pairs of congruent corresponding parts. ∠P ∠S, ∠Q ∠T,
−− −−
−−
−−
∠R ∠W, PQ ∠ST, PR ∠SW,
−−
−−
QR ∠TW
Example 2
Given: ABC DBC
∠FHE and ∠FHG are rt. .
∠FHE ∠FHG
m∠FHE = m∠FHG
(6x - 12)° = 90°
6x = 102
x = 17
When you write
a statement such
as ABC DEF,
you are also stating
which parts are
congruent.
B. Find m∠DBC.
53
Divide both sides by 6.
2a. 4
2b. 37°
Given: ∠YWX and ∠YWZ are right
−−
angles. YW bisects ∠XYZ. W is the
−− −− −−
midpoint of XZ. XY YZ.
EXAMPLE
Prove: XYW ZYW
3
Sum Thm.
Substitute values for m∠FHE
and m∠E.
<
1. ∠YWX and ∠YWZ are rt. .
(Given)
2. ∠YWX ∠YWZ (Rt. ∠ Thm.)
−−
3. YW bisects ∠XYZ. (Given)
4. ∠XYW ∠ZYW (Def. of ∠
bisector)
−−
5. W is mdpt. of XZ. (Given)
−− −−
6. XW ZW (Def. of mdpt.)
−− −−
7. YW YW (Reflex. Prop. of )
8. ∠X ∠Z (Third Thm.)
−− −−
9. XY YZ (Given)
10. XYW ZYW (Def. of )
Proving Triangles Congruent
Statements
Subtract 111.6 from both sides.
Corr. of are .
Def. of Trans. Prop. of =
£ä
+
,
*
Reasons
1. Given
2. Rt. ∠ Thm.
3. ∠PRQ ∠MRN
3. Vert. Thm.
4. ∠Q ∠N
4. Third Thm.
8. PQR MNR
È
2. ∠P ∠M
−−−
5. R is the mdpt. of PM.
−− −−−
6. PR MR
−− −−− −− −−
7. PQ MN; QR NR
5. Given
6. Def. of mdpt.
7. Given
8. Def. of −−
−−
3. Given: AD bisects BE.
−−
−−
BE bisects AD.
−− −−
AB DE, ∠A ∠D
Prove: ABC DEC
232
1. ∠P and ∠M are rt. Also available on transparency
INTERVENTION
Simplify.
Given: ∠P and ∠M are right angles.
−−
R is the midpoint of PM.
−− −−− −− −−
PQ MN, QR NR
Prove: PQR MNR
Proof:
9
7
Add 12 to both sides.
Given: ABC DEF
2a. Find the value of x.
xÎÂ
ÓÝÊÊÓ
2b. Find m∠F.
40.7°
Example 3
8
Substitute values for m∠FHE and m∠FHG.
m∠EFH + 111.6 = 180
m∠EFH = 68.4
∠GFH ∠EFH
m∠GFH = m∠EFH
m∠GFH = 68.4°
A. Find the value of x.
Rt. ∠ Thm.
Def. of m∠EFH + m∠FHE + m∠E = 180°
m∠EFH + 90 + 21.6 = 180
­ÓÝÊʣȮÂ
Def. of ⊥ lines
B Find m∠GFH.
{™°ÎÂ
­ÈÝÊʣӮ A Find the value of x.
Example 1
Ó£°ÈÂ
Chapter 4 Triangle Congruence
Questioning Strategies
E X AM P LE
1
2 Teach
• How does a triangle congruence ge07se_c04_0231_0237.indd 232
statement indicate corresponding
Guided Instruction
parts?
Show students how to name a polygon by
writing the vertices in consecutive order.
E X AM P LE 2
Discuss naming corresponding sides and
• What properties do you use to find
angles of congruent polygons, including
the measure of the angle?
those that overlap.
E X AM P LE
3
• Which proof statements could not
be placed in a different order?
232
Chapter 4
Reading Math Explain that the
everyday meaning of the
word consecutive is
ENGLISH
LANGUAGE
“following one another
LEARNERS
without interruption.”
5/8/06 12:36:34 PM
Through Visual Cues
Use two different-colored transparencies
of congruent triangles. Demonstrate on an
overhead projector or on a white board
that if two triangles are congruent, you can
slide one triangle exactly onto the other.
Have students identify the corresponding
sides and angles.
""
Overlapping Triangles
“With overlapping triangles, it helps me to redraw the triangles separately.
That way I can mark what I know about one triangle without getting confused
by the other one.”
£°x
£°x
Ó°£
£°Ó
£°Ó
ä°È
Cecelia Medina
Lamar High School
ä°Ç
Ê,,",
,/
In Example 4, students may write
JKL = MLK instead of JKL
MLK. Stress the difference
between the meanings of equal
and congruent.
Ó°£
£°Ó
£°Ó
ä°È
£
ä°Ç
£°Ç
£°Ç
Additional Examples
Example 4
EXAMPLE
4
Engineering Application
The bars that give structural support
to a roller coaster form triangles.
Since the angle measures and the
lengths of the corresponding sides are
the same, the triangles are congruent.
−− −− −− −−
Given: JK ⊥ KL, ML ⊥ KL, ∠KLJ ∠LKM,
−− −− −− −−
JK ML, JL MK
Prove: JKL MLK
Proof:
Statements
−− −− −−− −−
1. JK ⊥ KL, ML ⊥ KL
Reasons
1. Given
2. ∠JKL and ∠MLK are rt. .
2. Def. of ⊥ lines
3. ∠JKL ∠MLK
3. Rt. ∠ Thm.
4. ∠KLJ ∠LKM
4. Given
5. ∠KJL ∠LMK
−− −−− −− −−−
6. JK ML, JL MK
−− −−
7. KL LK
5. Third Thm.
8. JKL MLK
8. Def. of The diagonal bars across a gate
give it support. Since the angle
measures and the lengths of
the corresponding sides are
the same, the triangles are
congruent.
−−
−−
Given: PR and QT bisect each
−− −−
other. ∠PQS ∠RTS, QP RT
Prove: QPS TRS
+
,
-
*
/
−− −−
1. QP RT (Given)
2. ∠PQS ∠RTS (Given)
−−
−−
3. PR and QT bisect each other.
(Given)
−− −− −− −−
4. QS TS, PS RS (Def. of
bisector)
5. ∠QSP ∠TSR (Vert. Thm.)
6. ∠QPS ∠TRS (Third Thm.)
7. QPS TRS (Def. of )
6. Given
7. Reflex. Prop. of 4. Use the diagram to prove the following.
−−−
−− −−
−−− −− −−− −− −−−
Given: MK bisects JL. JL bisects MK. JK ML, JK ML
Prove: JKN LMN
Also available on transparency
THINK AND DISCUSS
INTERVENTION
1. A roof truss is a triangular structure that
supports a roof. How can you be sure that
two roof trusses are the same size and shape?
2. GET ORGANIZED Copy and complete
the graphic organizer. In each box, name the
congruent corresponding parts.
Questioning Strategies
E X A M P LE
̱*+,ÊɁÊ̱
˜}iÃ
4
• What all do you need to know for
this proof? Explain.
-ˆ`iÃ
Answers to Check It Out!
4-3 Congruent Triangles
3 Close
ge07se_c04_0231_0237.indd 233
Summarize
Review how to name corresponding angles
and sides of congruent polygons. Remind
students that to prove two triangles congruent, you must show all three pairs of
sides and all three pairs of angles are congruent.
233
3–4. See p. A15.
Answers to Think and Discuss
and INTERVENTION
Diagnose Before the Lesson
4-3 Warm Up, TE p. 231
1. Measure all the sides and all the
.
The trusses are the same size
and shape if all the corr. sides
and are .
12/2/05 7:11:16 PM
2. See p. A4.
Monitor During the Lesson
Check It Out! Exercises, SE pp. 231–233
Questioning Strategies, TE pp. 232–233
Assess After the Lesson
4-3 Lesson Quiz, TE p. 237
Alternative Assessment, TE p. 237
Lesson 4-3
233
4-3
4-3 Exercises
Exercises
KEYWORD: MG7 4-3
KEYWORD: MG7 Parent
GUIDED PRACTICE
Assignment Guide
Vocabulary Apply the vocabulary from this lesson to answer each question.
Assign Guided Practice exercises
as necessary.
1. An everyday meaning of corresponding is “matching.” How can this help you find the
corresponding parts of two triangles?
2. If ABC RST, what angle corresponds to ∠S? ∠B
If you finished Examples 1–2
Basic 13–18, 21, 23–25
Average 13–18, 21, 22–25
Advanced 13–18, 21, 22–25, 36
SEE EXAMPLE
1
Given: RST LMN. Identify the congruent corresponding parts.
−−
−−
−−
−−
LM
4. LN ? RT
5. ∠S ? ∠M
3. RS ?
−−−−
−−−−
−−−−
−−
−−
NM
6. TS ?
7. ∠L ? ∠R
8. ∠N ? ∠T
−−−−
−−−−
−−−−
2
Given: FGH JKL. Find each value.
p. 231
If you finished Examples 1–4
Basic 13–20, 23–26, 29,
31–34, 38–45
Average 13–26, 28–35, 38–45
Advanced 13–20, 22–28, 30–45
SEE EXAMPLE
9. KL 9
p. 232
10. x 32
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Homework Quick Check
3
p. 232
Quickly check key concepts.
Exercises: 13, 18, 19, 20, 24
−−
−−
11. Given: E is the midpoint of AC and BD.
−− −− −− −−
AB CD, AB CD
1. You find the and sides that are
in the same, or matching, places
in the 2 .
5. e. ?
−−−−
6. ∠AEB ∠CED
7. ABE CDE
p. 233
Reasons
a. Given
b. Alt. Int. Thm.
c. Given
d. Given
−− −− −− −−
e. AE CE, DE BE
f. Vert. Thm.
g. Def. of ?
−−−−
?
−−−−
3. c. ?
−−−−
4. d. ?
−−−−
5. Def. of mdpt.
1. a.
2. ∠ABE ∠CDE, ∠BAE ∠DCE
−− −−
3. AB CD
−−
−−
4. E is the mdpt. of AC and BD .
SEE EXAMPLE 4
Statements
−− −−
1. AB CD
Prove: ABE CDE
Proof:
Answers
Þ
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SEE EXAMPLE
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2. b.
6. f.
7. g.
?
−−−−
?
−−−−
12. Engineering The geodesic dome
shown is a 14-story building that
models Earth. Use the given
information to prove that the
triangles that make up the sphere
are congruent.
−− −− −− −− −−
Given: SU ST SR, TU TR,
∠UST ∠RST,
and ∠U ∠R
Prove: RTS UTS
U
S
-
T
1
R
,
12.
/
1. ∠UST ∠RST, ∠U ∠R (Given)
2. ∠STU
∠STR (Third −− −−
−−Thm.)
−−
3. SU
SR
(Given)
4.
ST
ST (Reflex. Prop. of )
−− −−
5. TU TR (Given) 6. RTS UTS (Def. of )
234
Chapter 4 Triangle Congruence
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PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
13–16
17–18
19
20
1
2
3
4
17. m∠C
18. y
31°
Extra Practice
ÞÊÊÇ
19
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Application Practice p. S31
−−−
19. Given: MP bisects ∠NMR. P is the midpoint of
−− −−− −−−
NR. MN MR, ∠N ∠R
Prove: MNP MRP
­ÝÊÊ££®Â
Visual Have students
draw and color code two
congruent polygons so
corresponding sides and angles are
the same color. Then have students
identify the corresponding parts.
*
Proof:
,
Statements
1. ∠N ∠R
−−−
2. MP bisects ∠NMR.
?
−−−−
?
−−−−
−−
5. P is the mdpt. of NR.
3. c.
4. d.
6. f. ?
−−−− −−−
−−−
7. MN MR
−−− −−−
8. MP MP
Reasons
?
−−−−
?
−−−−
3. Def. of ∠ bisector
1. a.
4. Third Thm.
?
−−−−
6. Def. of mdpt.
5. e.
?
−−−−
?
−−−−
9. Def. of 8. h.
20. Hobbies In a garden, triangular flower
beds are separated by straight rows of
grass as shown.
Inclusion In Exercises
19 and 20, have students
write an S or an A next to
each statement that states the congruence of a pair of sides or a pair
of angles. Then they can see if they
have listed enough parts to conclude
that the triangles are congruent.
a. Given
b. Given
c. ∠NMP ∠RMP
d. ∠NPM ∠RPM
e. Given
−− −−
f. PN PR
g. Given
h. Reflex. Prop. of 2. b.
7. g.
9. MNP MRP
A
Answers
B
Given: ∠ADC and ∠BCD are right angles.
−− −− −− −−
AC BD, AD BC
∠DAC ∠CBD
Prove: ADC BCD
GSR KPH;
SRG PHK;
RGS HKP
In Exercise 19, be sure students
understand that the order in which
vertices are written is important. In
line 3, if they name the angle ∠NMP,
then it must be congruent to ∠RMP.
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20. 1. ∠ ADC and ∠BCD are rt. .
(Given)
2. ∠ ADC ∠BCD (Rt. ∠ Thm.)
3. ∠DAC ∠CBD (Given)
4. ∠ ACD ∠BDC (Third Thm.)
−− −− −− −−
5. AC BD, AD BC (Given)
−− −−
6. DC DC (Reflex. Prop. of )
7. ADC BCD (Def. of )
E
D
C
21. For two triangles, the following
corresponding parts are given:
−− −− −− −− −− −−
GS KP, GR KH, SR PH,
∠S ∠P, ∠G ∠K, and ∠R ∠H.
Write three different congruence statements.
-
22. The two polygons in the diagram are congruent.
Complete the following congruence
statement for the polygons.
polygon R ? polygon V ?
−−−−
−−−−
Ê,,",
,/
Given: ABD CBD. Find each value.
Skills Practice p. S10
""
Given: Polygon CDEF polygon KLMN. Identify the congruent corresponding parts.
−−
−−
−−
−−
13. DE ? LM
14. KN ? CF
−−−−
−−−−
∠D
16. ∠L ?
15. ∠F ? ∠N
−−−−
−−−−
1
/
6
,
7
<
Possible answer: RVUTS VWXZY
8
9
Write and solve an equation for each of the following.
23. ABC DEF. AB = 2x - 10, and DE = x + 20. x = 30;
Find the value of x and AB.
AB = 50
°
2
24. JKL MNP. m∠L = (x + 10) , and m∠P = (2x 2 + 1)°. What is m∠L? 19°
25. Polygon ABCD polygon PQRS. BC = 6x + 5, and QR = 5x + 7.
Find the value of x and BC. x = 2; BC = 17
4-3 Congruent Triangles
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Lesson 4-3
235
Exercise 26 involves
proving triangles
congruent when they
are formed by folding paper. This
prepares students for the Multi-Step
Test Prep on page 238.
26. This problem will prepare you for the Multi-Step Test Prep on page 238.
Many origami models begin with a square piece of paper,
JKLM, that is folded along both diagonals to make the
−−
−−−
creases shown. JL and MK are perpendicular bisectors
of each other, and ∠NML ∠NKL.
−−
−−−
a. Explain how you know that KL and ML are congruent.
b. Prove NML NKL.
In Exercise 33, if students chose A, they
have made the measure of ∠Y equal to the measure of
∠C instead of ∠B.
28. Critical Thinking Draw two triangles that are not congruent but have an area of
4 cm 2 each.
Solution A is incorrect.
∠E ∠M, so m∠E = 46°.
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that the two given triangles are congruent?
ABC EFD
ABC DEF
ABC FDE
ABC FED
7
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8
C
32. MNP RST. What are the values of x and y?
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33. ABC XYZ. m∠A = 47.1°, and m∠C = 13.8°. Find m∠Y.
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42.9
119.1
34. MNR SPQ, NL = 18, SP = 33, SR = 10, RQ = 24,
and QP = 30. What is the perimeter of MNR?
79
85
28. Possible answer:
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-
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,
Chapter 4 Triangle Congruence
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30. Write About It Given the diagram of the
triangles, is there enough information to prove
that HKL is congruent to YWX? Explain.
−−
−−
1. BD ⊥ AC (Given)
2. ∠ ADB and ∠CDB
are rt. (Def. of ⊥)
3. ∠ ADB ∠CDB
(Rt. ∠ Thm.)
−−
4. BD bisects ∠ ABC. (Given)
5. ∠ ABD ∠CBD (Def. of bisect)
6. ∠ A ∠C (Third Thm.)
−− −−
7. AB CB (Given)
−− −−
8. BD DB (Reflex. Prop. of )
−−
9. D is the mdpt. of AC. (Given)
−− −−
10. AD CD (Def. of mdpt.)
11. ABD CBD (Def of )
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/////ERROR ANALYSIS/////
29.
−− −−
26a. KL ML by the def. of a square.
b.1. JKLM is a square. (Given)
−− −−
2. KL ML (Def. of a square)
−−
−−
3. JL and MK are ⊥ bisectors of
each other. (Given)
−− −−
4. MN KN (Def. of bisect)
−− −−
5. NL NL (Reflex. Prop. of )
6. ∠MNL and ∠KNL are rt. .
(Def. of ⊥)
7. ∠MNL ∠KNL (Rt. ∠ Thm.)
8. ∠NML ∠NKL (Given)
9. ∠NLM ∠NLK (Third Thm.)
10. NML NKL (Def. of )
A
27. Draw a diagram and then write a proof.
−− −− −−
−− −−
−−
Given: BD ⊥ AC. D is the midpoint of AC. AB CB, and BD bisects ∠ABC.
Prove: ABD CBD
Answers
27.
34
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CHALLENGE AND EXTEND
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35. Multi-Step Given that the perimeter of TUVW is 149 units,
find the value of x. Is TUV TWV? Explain.
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36. Multi-Step Polygon ABCD polygon EFGH. ∠A is a right angle.
m∠E = (y 2 - 10)°, and m∠H = (2y 2 - 132)°. Find m∠D. 68°
−− −−
,
37. Given: RS RT, ∠S ∠T
Prove: RST RTS
/
-
SPIRAL REVIEW
Two number cubes are rolled. Find the probability of each outcome.
(Previous course)
_
_
39. The sum of the numbers rolled is 5. 1
38. Both numbers rolled are even. 1
4
Visual For Exercise 37,
suggest that students draw
two separate triangles.
1
9
35. x = 5.5; yes; UV = WV = 41.5,
and UT = WT = 33. TV = TV by
the Reflex. Prop. of =. It is given
that ∠VWT ∠VUT and ∠WTV
∠UTV. ∠WVT ∠UVT by the
Third Thm. Thus TUV TWV by the def. of .
−− −−
37. 1. RS RT; ∠S ∠T (Given)
−− −−
2. ST TS (Reflex. Prop. of )
3. ∠T ∠S (Sym. Prop. of )
4. ∠R ∠R (Reflex. Prop of )
5. RST RTS (Def. of )
Classify each angle by its measure. (Lesson 1-3)
40. m∠DOC = 40° acute
41. m∠BOA = 90° rt.
42. m∠COA = 140° obtuse
Find each angle measure. (Lesson 4-2)
43. ∠Q 72°
44. ∠P 74°
Have students explain why the order
of vertices is important in congruence statements.
+
45. ∠QRS 146°
Journal
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explains what it means for two realworld objects to be congruent. They
should identify the corresponding
angles and sides of each.
KEYWORD: MG7 Career
Q:
A:
What math classes did you take in high school?
Q:
A:
What kind of degree or certification will you receive?
Q:
A:
How do you use math in your hands-on training?
Q:
A:
Jordan Carter
Emergency Medical
Services Program
Algebra 1 and 2, Geometry, Precalculus
I will receive an associate’s degree in applied science.
Then I will take an exam to be certified as an EMT
or paramedic.
4-3
I calculate dosages based on body weight and age. I also
calculate drug doses in milligrams per kilogram per hour or
set up an IV drip to deliver medications at the correct rate.
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congruent corresponding part.
−−
−−
2. NO ?
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What are your future career plans?
4-3 Congruent Triangles
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1. ∠ A ∠E (Given)
−−
−−
2. C is mdpt. of BD and AE.
(Given)
−− −− −− −−
3. AC EC; BC DC (Def. of
mdpt.)
−− −−
4. AB ED (Given)
5. ∠ ACB ∠ECD (Vert. Thm.)
6. ∠B ∠D (Third Thm.)
7. ABC EDC (Def. of )
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Lesson 4-3
237
SECTION 4A
SECTION
4A
Triangles and Congruence
Organizer
Origami Origami is the Japanese art of paper folding.
The Japanese word origami literally means “fold paper.”
This ancient art form relies on properties of geometry to
produce fascinating and beautiful shapes.
Each of the figures shows a step in making an origami
swan from a square piece of paper. The final figure shows
the creases of an origami swan that has been unfolded.
Objective: Assess students’
GI
ability to apply concepts and skills
in Lessons 4-1 through 4-3 in a
real-world format.
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Step 1
Online Edition
Step 2
Step 3
Resources
Geometry Assessments
www.mathtekstoolkit.org
Problem
Text Reference
1
Lesson 4-1
2
Lesson 4-2
3
Lesson 4-3
Fold the paper in half
diagonally and crease it.
Turn it over.
Fold corners A and C
to the center line and
crease. Turn it over.
Fold in half along the
−−
center crease so that DE
−−
and DF are together.
Step 4
Step 5
Step 6
É
Answers
−−
2. m∠EBD = 45° (DB bisects rt.
−−
∠ ABC.) m∠BDE = 22.5° (DE
bisects ∠ ADB.) m∠DEB = 112.5°
( Sum Thm.)
−−
3. 1. DB bisects ∠ ABC and ∠EDF.
(Given)
2. ∠EBD ∠FBD; ∠EDB ∠FDB (Def. of ∠ bisector)
3. ∠DEB ∠DFB (Third ∠ Thm.)
−− −− −− −−
4. BE BF; DE DF (Given)
−− −−
5. DB DB (Reflex. Prop. of )
6. EDB FDB (Def. of )
Fold the narrow point
upward at a 90° angle
and crease. Push in the
fold so that the neck is
inside the body.
É
Fold the tip downward
and crease. Push in the
fold so that the head is
inside the neck.
isosc. ; 1. Use the fact that ABCD is a square
to classify ABD by its side
rt. lengths and by its angle measures.
−−
−−
2. DB bisects ∠ABC and ∠ADC. DE
bisects ∠ADB. Find the measures
of the angles in EDB. Explain
how you found the measures.
−−
3. Given that DB bisects ∠ABC and
−− −−
−− −−
∠EDF, BE BF, and DE DF,
prove that EDB FDB.
238
É
Fold up the flap to
form the wing.
Chapter 4 Triangle Congruence
INTERVENTION
Scaffolding Questions
ge07se_c04_0238_0239.indd 238
KEYWORD: MG7 Resources
238
Chapter 4
1. Explain why ABD cannot be an equilateral triangle. The 3 sides cannot all have
equal lengths since DB 2 = AB 2 + AD 2 by
the Pyth. Thm.
−−
2. Suppose DE trisects ∠ ADB so that
1
m∠EDB = __
m∠ ADB. What is
3
m∠ AED? 60°
−− −− −− −−
3. Suppose that AE CF, AD CD,
−− −−
DE DF, and that ∠ A and ∠C are right
angles. Is there enough information to
prove that ADE CDF ? If not, what
additional information is needed? No; you
need to know ∠ ADE ∠CDF or ∠ AED ∠CFD.
Extension
Draw a large scalene ABC. Then construct
−−
equilateral ABC outward from side AB.
Similarly construct equilateral BCA and
ACB. Bisect each angle of the three equilateral triangles. Where the bisectors meet
label the points D, E, and F. How would you
classify DEF ? equilateral
8/15/05 9:49:08 AM
SECTION 4A
SECTION
Quiz for Lessons 4-1 Through 4-3
4-1 Classifying Triangles
ÎäÂ
Classify each triangle by its angle measures.
1. ACD rt.
4A
ÎäÂ
2. ABD equiangular 3. ADE obtuse
ÈäÂ
ÈäÂ
Classify each triangle by its side lengths.
4. PQR isosc.
5. PRS equil.
6. PQS scalene
Objective: Assess students’
*
n°Ç
+
4-2 Angle Relationships in Triangles
Organizer
mastery of concepts and skills in
Lessons 4-1 through 4-3.
x
-
,
Find each angle measure.
7. m∠M
8. m∠ABC
51°
125°
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Section 4A Quiz
,
9. A carpenter built a triangular support structure for a roof. Two of the
angles of the structure measure 37° and 55°. Find the measure of ∠RTP,
the angle formed by the roof of the house and the roof of the patio. 92°
xxÂ
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-
/ *
4-3 Congruent Triangles
Given: JKL DEF. Identify the congruent corresponding parts.
−−
−−
−−
−−
EF
10. KL ?
11. DF ? JL
12. ∠K ? ∠E
−−−−
−−−−
−−−−
Given: PQR STU. Find each value.
14. PQ 9
,
+
15. y 23
ӓÊÊ£
{ÈÂ
−− −− −− −−
AB CD, AC BD,
16. Given: AB
CD,
−− −− −− −−
AC ⊥ CD, DB ⊥ AB
Prove: ACD DBA
INTERVENTION
13. ∠F Resources
1
/
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*
∠L
?
−−−−
Ready to Go On?
Intervention and
Enrichment Worksheets
ΓÊÊÓ
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Ready to Go On? CD-ROM
-
Ready to Go On? Online
Proof:
Statements
Reasons
CD
1. AB
1. a.
2. ∠BAD ∠CDA
−− −− −− −−
3. AC ⊥ CD, DB ⊥ AB
2. b.
4. ∠ACD and ∠DBA are rt. ?
−−−−−
?
6. f.
−− −− −−
−−−−−−−
7. AB CD, AC BD
5. e.
?
−−−−−
9. ACD DBA
8. h.
?
−−−−−
?
−−−−−
?
3. c.
−−−−−
4. d.
?
−−−−−
5. Rt. ∠ Thm.
a. Given
b. Alt. Int. Thm.
c. Given
d. Def. of ⊥
e. ∠ACD ∠DBA
f. ∠CAD ∠BDA −− −−
g. Given
h. AD DA
i. Def. of 6. Third Thm.
?
−−−−−
8. Reflex Prop. of 7. g.
?
−−−−−
9. i .
Ready to Go On?
NO
ge07se_c04_0238_0239.indd 239
READY
Ready to Go On?
Intervention
Lesson 4-1
YES
Diagnose and Prescribe
INTERVENE
TO
Worksheets
4-1 Intervention
239
ENRICH
5/8/06 12:42:16 PM
GO ON? Intervention, Section 4A
CD-ROM
Online
Activity 4-1
Lesson 4-2
4-2 Intervention
Activity 4-2
Lesson 4-3
4-3 Intervention
Activity 4-3
Diagnose and
Prescribe Online
READY TO GO ON?
Enrichment, Section 4A
Worksheets
CD-ROM
Online
Ready to Go On?
239
SECTION
4B Proving Triangle Congruence
One-Minute Section Planner
Lesson
Lab Resources
4-4 Geometry Lab Explore SSS and SAS Triangle Congruence
•
□
Discover shortcuts for proving triangles are congruent.
SAT-10 ✔ NAEP
ACT
SAT
SAT Subject Tests
□
□
□
□
Geometry Lab Activities
4-4 Lab Recording Sheet
Materials
Required
straws, string, paper clip, and
protractor (MK)
Optional
envelopes with three
descriptions of parts of a
triangle, ruler, protractor (MK)
Lesson 4-4 Triangle Congruence: SSS and SAS
•
•
□
Apply SSS and SAS to construct triangles and solve problems.
Prove triangles congruent by using SSS and SAS.
SAT-10 ✔ NAEP
ACT
SAT
SAT Subject Tests
□
□
□
Geometry Lab Activities
4-4 Geometry Lab
□
Required
straightedge, compass (MK),
geometry software
Optional
magazine pictures showing
triangle congruence, ruler,
protractor (MK)
4-5 Technology Lab Predict Other Triangle Congruence
Relationships
•
□
Technology Lab Activities
4-5 Lab Recording Sheet
Required
geometry software
Use geometry software to explore triangle congruence relationships.
SAT-10
NAEP
ACT
✔ SAT
✔ SAT Subject Tests
□
□
□
□
Lesson 4-5 Triangle Congruence: ASA, AAS, and HL
• Apply ASA, AAS, and HL to construct triangles and solve problems.
• Prove triangles congruent by using ASA, AAS, and HL.
✔ NAEP □ ACT
□ SAT-10 □
□ SAT □ SAT Subject Tests
Lesson 4-6 Triangle Congruence: CPCTC
Lesson 4-7 Introduction to Coordinate Proof
Position figures in the coordinate plane for use in coordinate proofs.
Prove geometric concepts by using coordinate proofs.
SAT Subject Tests
✔ SAT-10 ✔ NAEP ✔ ACT
✔ SAT
□
□
□
□
Lesson 4-8 Isosceles and Equilateral Triangles
•
•
□
Prove theorems about isosceles and equilateral triangles.
Apply properties of isosceles and equilateral triangles.
SAT-10 ✔ NAEP
✔ ACT
✔ SAT
✔ SAT Subject Tests
□
□
□
straightedge, compass (MK)
Optional
map of the area that contains
your school, paper and scissors
Optional
• Use CPCTC to prove parts of triangles are congruent.
✔ NAEP □ ACT
□ SAT-10 □
□ SAT □ SAT Subject Tests
•
•
□
Required
□
designs by M. C. Escher
Required
graph paper
Optional
scissors
Required
graph paper
Optional
different examples of isosceles
and equilateral triangles, poster
board, ruler (MK)
Extension Proving Constructions Valid
• Use congruent triangles to prove constructions valid.
✔ NAEP □
✔ ACT
□ SAT-10 □
□ SAT □ SAT Subject Tests
MK = Manipulatives Kit
240A
Chapter 4
Section Overview
Triangle Congruence: SSS, SAS, ASA, AAS, HL, CPCTC Lessons 4-4, 4-5, 4-6
Triangles have special properties that allow you to use shortcuts for proving
triangles congruent.
−−
AB −−
BC −−
CA SSS
−−
DE
−−
EF
−−
FD
SAS
AAS
∠ A ∠D
−− −−
AC DF
∠C ∠F
ABC DEF
ABC DEF
ABC DEF
HL
−−
EF
−−
DE
−−
BC −−
AB ∠ A ∠D
∠B ∠E
−− −−
BC EF
ASA
−− −−
AB DE
∠B ∠E
−− −−
BC EF
ABC DEF
ABC DEF
CPCTC = Corresponding Parts of Congruent Triangles are Congruent.
Once you know that two triangles are congruent, you know that all corresponding parts are congruent.
Introduction to Coordinate Proof
Lesson 4-7
Coordinate proof is a style of proof that uses coordinate geometry and algebra.
Strategies for Positioning Figures in the
Coordinate Plane:
• Use the origin as a vertex, keeping the
figure in Quadrant 1.
• Center the figure at the origin.
• Center a side of the figure at the origin.
• Use one or both axes as sides of the figure.
Coordinate proof uses coordinates,
the Midpoint Formula, the Pythagorean
Theorem, and/or the Distance Formula to
prove conjectures.
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Isosceles and Equilateral Triangles
Lesson 4-8
Isosceles and equilateral triangles frequently appear in other figures, so
knowing the properties of isosceles and equilateral triangles is very useful.
Equilateral Equiangular Isosceles RST
−− −− −−
AB BC CA ⇔ ∠ A ∠B ∠C
−− −− −−
AB BC CA if and only if ∠ A ∠B ∠C.
−− −−
RS ST ⇔ ∠R ∠T
−− −−
RS ST if and only if ∠R ∠T.
-
,
/
240B
4-4
Organizer
Explore SSS and SAS
Triangle Congruence
Use with Lesson 4-4
Pacing:
Traditional 1 day
1
Block __
day
2
Objective: Discover shortcuts for
proving triangles are congruent.
Use with Lesson 4-4
In Lesson 4-3, you used the definition of congruent triangles to prove
triangles congruent. To use the definition, you need to prove that all
three pairs of corresponding sides and all three pairs of corresponding
angles are congruent.
In this lab, you will discover some shortcuts for proving triangles congruent.
Materials: straws, string,
GI
paperclip, protractor
<D
@<I
Activity 1
Online Edition
1 Measure and cut six pieces from the straws:
two that are 2 inches long, two that are
4 inches long, and two that are 5 inches long.
Countdown to
Testing Week 8
2 Cut two pieces of string that are each about
20 inches long.
Resources
3 Thread one piece of each size of straw onto
a piece of string. Tie the ends of the string
together so that the pieces of straw form a
triangle.
Geometry Lab Activities
4-4 Lab Recording Sheet
4 Using the remaining pieces, try to make
another triangle with the same side lengths
that is not congruent to the first triangle.
Teach
Discuss
Discuss with students their attempts
to make a second triangle using the
same side lengths as the first triangle. Lead them to the conjecture
that two triangles with three pairs of
congruent corresponding sides are
congruent.
2. It is not possible. Once the lengths of the 3 straws
are determined, only 1 can be formed.
Try This
1. Repeat Activity 1 using side lengths of your choice. Are your results the same? yes
2. Do you think it is possible to make two triangles that have the same side lengths
but that are not congruent? Why or why not?
In the second activity, discuss with
students which combinations of
sides and angles form congruent triangles. Lead them to the conclusion
that the angle must be “between”
the two sides. If two corresponding
sides and their included angles are
congruent, then the triangles are
congruent.
3. How does your answer to Problem 2 provide a shortcut for proving triangles
congruent?
4. Complete the following conjecture based on your results. Two triangles are
?
.
congruent if
−−−−−−−−−−−−− three sides of 1 are to 3 sides of the other 240
Chapter 4 Triangle Congruence
ge07se_c04_0240_0241.indd 240
KEYWORD: MG7 Resources
240
Chapter 4
3. To prove that 2 are , check to see if the 3 pairs
of corr. sides are .
8/15/05 9:52:07 AM
Alternative Approach
Activity 2
Place students in small groups.
Give each student in the group an
envelope containing three descriptions of parts of a triangle. For example, AB is 4 in., m∠ A = 32°, and
m∠B = 48°. At least one envelope
should have the same characteristics
as another envelope given to the
group. The group should determine
how many of the triangles are
congruent and why.
1 Measure and cut two pieces from the straws:
one that is 4 inches long and one that is
5 inches long.
2 Use a protractor to help you bend a paper
clip to form a 30° angle.
3 Place the pieces of straw on the sides of the
30° angle. The straws will form two sides of
your triangle.
4 Without changing the angle formed by the
paper clip, use a piece of straw to make a
third side for your triangle, cutting it to fit
as necessary. Use additional paper clips
or string to hold the straws together in a
triangle.
Close
Key Concept
You do not need all six pairs of congruent corresponding parts to prove
triangles congruent. If you know
three pairs of corresponding sides
are congruent (SSS) or two pairs of
corresponding sides and the included angles are congruent (SAS),
then you know the triangles are
congruent.
6. No; once 2 side lengths and the included ∠ measure are
determined, only 1 length is possible for the remaining side.
Try This
7. To prove that 2 are , check to see if there are 2 pairs
of corr. sides and that their included are .
5. Repeat Activity 2 using side lengths and an angle measure of your choice.
Are your results the same? yes
6. Suppose you know two side lengths of a triangle and the measure of the angle
between these sides. Can the length of the third side be any measure? Explain.
Assessment
Journal Have students compare and
contrast the SSS and SAS postulates
and support their answers with a
sketch that illustrates each postulate.
7. How does your answer to Problem 6 provide a shortcut for proving triangles
congruent?
8. Use the two given sides and the given angle from Activity 2 to form a triangle
that is not congruent to the triangle you formed. (Hint: One of the given sides
does not have to be adjacent to the given angle.) Check students’ work.
9. Complete the following conjecture based on your results.
Two triangles are congruent if
?
.
−−−−−−−−−−−−−
2 sides and the included ∠ of
1 are to 2 sides and the
included ∠ of the other 4- 4 Geometry Lab
ge07se_c04_0240_0241.indd 241
241
8/15/05 9:52:13 AM
4-4 Geometry Lab
241
4-4
Organizer
4-4
Pacing: Traditional 2 days
Triangle Congruence:
SSS and SAS
Block 1 day
Objectives: Apply SSS and SAS
to construct triangles and solve
problems.
Prove triangles congruent by using
SSS and SAS.
GI
In Geometry Lab Activities
<D
@<I
Prove triangles congruent
by using SSS and SAS.
Vocabulary
triangle rigidity
included angle
Geometry Lab
Who uses this?
Engineers used the property of
triangle rigidity to design the
internal support for the Statue
of Liberty and to build bridges,
towers, and other structures.
(See Example 2.)
Objectives
Apply SSS and SAS to
construct triangles and
to solve problems.
Online Edition
In Lesson 4-3, you proved triangles congruent
by showing that all six pairs of corresponding
parts were congruent.
The property of triangle rigidity gives you a shortcut
for proving two triangles congruent. It states that if
the side lengths of a triangle are given, the triangle
can have only one shape.
Tutorial Videos
Countdown to
Testing Week 8
For example, you only need to know that two triangles
have three pairs of congruent corresponding sides.
This can be expressed as the following postulate.
Postulate 4-4-1
Warm Up
Side-Side-Side (SSS) Congruence
POSTULATE

1. Name the angle formed by AB
. Possible answer: ∠ A
and AC
If three sides of one
triangle are congruent
to three sides of
another triangle, then
the triangles
are congruent.
2. Name the three sides of
−− −− −−
ABC. AB, AC, BC
3. QRS LMN. Name all
pairs of congruent corre−− −−
sponding parts. QR LM,
−− −− −− −−
RS MN, QS LN, ∠Q ∠L,
∠R ∠M, ∠S ∠N
EXAMPLE
1
{ÊV“
ÇÊV“
ÈÊV“
Adjacent triangles
share a side, so
you can apply the
Reflexive Property
to get a pair of
congruent parts.
{ÊV“
ABC FDE
ÈÊV“
ÇÊV“
+
Using SSS to Prove Triangle Congruence
*
−− −−
−− −−
It is given that AB CD and−−BC −−DA.
By the Reflex. Prop. of , AC CA.
So ABC CDA by SSS.
An included angle is an angle formed by two
adjacent sides of a polygon. ∠B is the included
−−
−−
angle between sides AB and BC.
,
-
1. Use SSS to explain why
ABC CDA.
Q: Why did the greeting card come
after your birthday?
242
CONCLUSION
Use SSS to explain why PQR PSR.
−− −−
−− −−
It is given that PQ PS and that QR SR. By
−− −−
the Reflexive Property of Congruence, PR PR.
Therefore PQR PSR by SSS.
Also available on transparency
A: Postulate!
HYPOTHESIS
Chapter 4 Triangle Congruence
1 Introduce
ge07se_c04_0242_0249.indd 242
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242
Chapter 4
%XPLAIN HOWYOUCOULDUSETHEABOVE
CONJECTURE TO SHOW THAT N103 N203
0
Motivate
Use pictures from magazines to show students
that triangle congruence is important in designing
and building structures. Triangles can be proved
congruent without using all three pairs of angles
and all three pairs of sides. Explain to students
that this lesson will show them how to prove triangles congruent using three pairs of congruent
corresponding parts.
Explorations and answers are provided in the
Explorations binder.
5/8/06 12:47:39 PM
It can also be shown that only two pairs of congruent corresponding sides are
needed to prove the congruence of two triangles if the included angles are
also congruent.
Postulate 4-4-2
HYPOTHESIS
If two sides and the included
angle of one triangle are
congruent to two sides
and the included angle of
another triangle, then the
triangles are congruent.
2
ABC EFD
Students may choose the wrong
angle when SAS is used to prove
triangles congruent. Explain that the
angle must be formed by the sides.
The included angle is named by the
letter the segments share.
Transformations Lead
students to recognize
when reflection is modeled
in the examples. You can do this
with a mirror (MK) or Mira.
Engineering Application
The figure shows part of the support
structure of the Statue of Liberty.
Use SAS to explain why
KPN LPM.
K
−− −−
It is given that KP LP
−− −−−
and that NP MP.
By the Vertical Angles
Theorem, ∠KPN ∠LPM.
N
Therefore KPN LPM
by SAS.
The letters SAS
are written in that
order because the
congruent angles
must be between
pairs of congruent
corresponding sides.
CONCLUSION
Ê,,",
,/
Side-Angle-Side (SAS) Congruence
POSTULATE
EXAMPLE
""
Additional Examples
M
Example 1
P
Use SSS to explain why
ABC DBC.
L
2. Use SAS to explain
why ABC DBC.
−− −−
2. It is given that BA BD −−
and ∠ABC
−− ∠DBC.
By the Reflex. Prop. of , BC BC.
So ABC DBC by SAS.
−− −−
It is given that AC DC and
−− −−
that AB DB. By the Reflex.
−− −−
Prop. of , BC BC . Therefore
ABC DBC by SSS.
The SAS Postulate guarantees that if you are given the lengths of two sides and
the measure of the included angle, you can construct one and only one triangle.
Construction Congruent Triangles Using SAS
Use a straightedge to draw two segments and one angle,
or copy the given segments and angle.
Example 2
The diagram shows part of the
support structure for a tower.
Use SAS to explain why XYZ VWZ.
8
9
<
7
−−
Construct AB congruent to one
of the segments.
Construct ∠A congruent to
the given angle.
6
−− −−
It is given that XZ VZ and
−− −−
that YZ WZ. By the Vert. Thm. ∠XZY ∠VZW. Therefore
XYZ VWZ by SAS.
−−
Construct AC congruent to
−−
the other segment. Draw CB
to complete ABC.
Also available on transparency
4-4 Triangle Congruence: SSS and SAS
INTERVENTION
2 Teach
Questioning Strategies
ge07se_c04_0242_0249.indd 243
12/5/05 6:58:54 AM
E X A M P LE
Guided Instruction
Review with students how to write congruence statements based on corresponding
parts. Explain that the SSS and the SAS
congruence postulates are shortcuts to
verifying all six corresponding parts congruent. Draw triangles with the following
measures: one side 7 cm, one side 10 cm,
and a 40° angle. Explain that the triangles
are not necessarily congruent. The triangles drawn with the 40° angle included
between the given sides are congruent.
243
Through Modeling
Introduce the SSS postulate with the
following activity. Have the students draw
three line segments of given lengths. Using
one of the three lengths as a base and one
endpoint of the base as center, draw an
arc with a radius equal to a second length.
Draw an arc with a radius equal to the
third length, using the other endpoint as
center. Join the endpoints with the intersection of the arcs. Have students compare
their triangles and make a conjecture.
1
• What do the tick marks on the
triangles show? What additional
information do you need to prove
the triangles congruent by SSS?
E X A M P LE
2
• Is enough information given to
prove the triangles congruent by
SAS? What additional information
do you need?
Lesson 4-4
243
EXAMPLE
3
Additional Examples
A UVW YXW, x = 3
Example 3
A. MNO PQR, when x = 5.
+
x
"
È
ÝÊÊÓ
Ý
,
ÎÝÊʙ
B DEF JGH, y = 7
*
JG = 2y + 1
= 2 (7) + 1
= 15
GH = y 2 - 4y + 3
= (7) 2 - 4 (7) + 3
= 24
m∠G = 12y + 42
= 12 (7) + 42
= 126°
−− −− −− −−−
DE JG. EF GH, and ∠E ∠G.
So DEF JGH by SAS.
MNO PQR by SSS.
B. STU VWX, when y = 4
-
1
ÞÊÊÎ
7
££
Ç ™ÓÂ
6
ÓÞÊÊÎ
­ÓäÞÊÊ£Ó®
 8
/
3. DA
−− = 13,
−− = DC
so DA DC by def.
of . m∠ADB =
m∠CDB = 32°, so
∠ADB ∠CDB by
−− −−
def. of . DB DB
by the Reflex. Prop.
of . Therefore
ADB CDB
by SAS.
STU VWX by SAS.
Example 4
−− −− −− −−
Given: BC AD, BC AD
Prove: ABD CDB
−− −−
1. BC AD (Given)
2. ∠CBD ∠ADB (Alt. Int. Thm.)
−− −−
3. BC AD (Given)
−− −−
4. BD BD (Reflex. Prop. of )
5. ABD CDB (SAS Steps 3,
2, 4)
Also available on transparency
INTERVENTION
Questioning Strategies
E X AM P LE
3
• How do you find the lengths of the
sides of both triangles?
E X AM P LE
EXAMPLE
−− −−
1. QR QS (Given)
 bisects ∠RQS.
2. QP
(Given)
3. ∠RQP ∠SQP
(Def.
−−of bisector)
−−
4. QP QP (Reflex.
Prop. of )
5. RQP SQP
(SAS Steps 1, 3, 4)
ÎÝÊÊx
Ý
7
Î
<
ÝÊÊ£
9
£x
£ÓÈÂ
ÓÞÊÊ£
Ó{
ÞÓÊÊ{ÞÊÎ
­£ÓÞÊÊ{Ó®Â
ÎÌÊÊ£
ÎÓÂ
Ê ÊÓÌÊÓÊ
ÊÂ
{ÌÊÊÎ
4
Proving Triangles Congruent
−− −−
Given: m, EG HF
Prove: EGF HFG
Proof:
Statements
−− −−
1. EG HF
1. Given
2. m
2. Given
3. ∠EGF ∠HFG
−− −−
4. FG GF
3. Alt. Int. Thm.
5. EGF HFG
5. SAS Steps 1, 3, 4
Ű
“
Reasons
4. Reflex Prop. of −− −−
bisects ∠RQS. QR QS
4. Given: QP
Prove: RQP SQP
+
,
244
8
Ó
{
3. Show that ADB CDB
when t = 4.
4
• How do parallel lines help you
verify congruent angles?
1
ZY = x - 1
=3-1=2
XZ = x = 3
XY = 3x - 5
6
= 3 (3) - 5 = 4
−− −− −−− −−
−−− −−
UV YX. VW XZ, and UW YZ.
So UVW YXZ by SSS.
Show that the triangles are
congruent for the given value
of the variable.
Ç
Verifying Triangle Congruence
Show that the triangles are congruent
for the given value of the variable.
*
-
Chapter 4 Triangle Congruence
3 Close
ge07se_c04_0242_0249.indd 244
Summarize
Review the SSS and SAS postulates for
proving triangles congruent, and give
examples of each. Remind students of the
importance of the order of the letters in a
congruence statement.
and INTERVENTION
Diagnose Before the Lesson
4-4 Warm Up, TE p. 242
Monitor During the Lesson
Check It Out! Exercises, SE pp. 242–244
Questioning Strategies, TE pp. 243–244
Assess After the Lesson
4-4 Lesson Quiz, TE p. 249
Alternative Assessment, TE p. 249
244
Chapter 4
5/8/06 1:12:02 PM
Answers to Think and Discuss
THINK AND DISCUSS
1. Describe three ways you could
prove that ABC DEF.
2. Explain why the SSS and SAS
Postulates are shortcuts for
proving triangles congruent.
3. GET ORGANIZED Copy and
complete the graphic organizer.
Use it to compare the SSS and
SAS postulates.
4-4
1. Show that all six pairs of corr.
parts are ; SSS; SAS
2. The SSS and SAS Post. are methods for proving without
having to prove the congruence
of all 6 corr. parts.
---
3. See p. A4.
--
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Exercises
4-4 Exercises
KEYWORD: MG7 4-4
KEYWORD: MG7 Parent
GUIDED PRACTICE
Assignment Guide
−−
−−
1. Vocabulary In RST which angle is the included angle of sides ST and TR? ∠T
SEE EXAMPLE
1
2. ABD CDB
p. 242
Assign Guided Practice exercises
as necessary.
Use SSS to explain why the triangles in each pair are congruent.
3. MNP MQP
If you finished Examples
Basic 8–10, 14–18,
Average 8–10, 14–18,
Advanced 8–10, 14–18,
*
SEE EXAMPLE
2
p. 243
p. 244
3
If you finished Examples 1–4
Basic 8–18, 21, 23, 25, 26,
28–32, 37–45
Average 8–19, 21, 22–31, 33,
36–44
Advanced 8–14, 19–44
+
H
J
4. Sailing Signal flags are used to communicate
messages when radio silence is required.
The Zulu signal flag means, “I require a tug.”
GJ = GH = GL = GK = 20 in. Use SAS to
explain why JGK LGH.
G
Homework Quick Check
L
K
SEE EXAMPLE
Quickly check key concepts.
Exercises: 8, 10, 12, 13, 14, 25
Show that the triangles are congruent for the given value of the variable.
5. GHJ IHJ, x = 4
Î
ÎÝÊʙ
x
ÓÝÊÊÎ
1–2
27
24, 27
24–27
6. RST TUR, x = 18
,
When x = 4,
HI = GH = 3,
and
= GJ = 5.
−− IJ−−
HJ HJ by the
Reflex. Prop. of .
Therefore GHJ IHJ by SSS.
1
È£
-
ÎÈÂ
ÓÝÂ
{ÝÊÊ££
/
When x = 18, RS = UT = 61,
and
= m∠UTR = 36°.
−− m∠SRT
−−
RT TR by the Reflex. Prop. of .
So RST TUR by SAS.
4-4 Triangle Congruence: SSS and SAS
Critical Thinking In
Exercises 5 and 6, have
students consider how
they can find the value of the variable if the triangles are given as
congruent.
245
Answers
ge07se_c04_0242_0249.indd 245
−− −−
−− −−
2. It is given that DA BC and AB CD.
−− −−
BD DB by the Reflex. Prop. of .
Thus ABD CDB by SSS.
−− −−−
3. It is given that MN MQ and
−− −− −− −−
NP QP. MP MP by the Reflex. Prop.
of . Thus MNP MQP
by SSS.
−− −−
−− −−
4. It is given that JG LG and GK GH.
∠ JGK ∠LGH by the Vert. Thm. So
JGK LGH by SAS.
5/8/06 12:48:27 PM
KEYWORD: MG7 Resources
Lesson 4-4
245
Visual In Exercises 11
and 12, have students
label the congruent corresponding sides with S and the congruent corresponding angles with A.
SEE EXAMPLE 4
p. 244
−− −−−
7. Given: JK ML, ∠JKL ∠MLK
Answers
10. It is given that ∠C and ∠B are rt.
−− −−
and EC DB. ∠C ∠B by the
−− −−
Rt. ∠ Thm. CB BC by the
Reflex. Prop. of . So ECB DBC by SAS.
Proof:
Statements
−− −−−
1. JK ML
8. It is given that BC = ED = 4 in.
and BD = EC = 3 in. So by the
−− −−
−−
def. of , BC ED, and BD −− −− −−
EC. CD DC by the Reflex.
Prop. of . Thus BCD EDC by SSS.
−− −−
9. It is given that KJ LJ and
−− −− −− −−
GK GL. GJ GJ by the Reflex.
Prop. of . So GJK GJL by
SSS.
Prove: JKL MLK
2. b. ?
−−−−−−
−−
3. KL LK
Reasons
a. Given
b. ∠JKL ∠MLK
c. Reflex. Prop. of d. SAS Steps 1, 2, 3
1. a. ?
−−−−
2. Given
3. c.
4. JKL MLK
4. d.
?
−−−−
?
−−−−
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
8–9
10
11–12
13
1
2
3
4
Use SSS to explain why the triangles in each pair are congruent.
8. BCD EDC
{ʈ˜°
9. GJK GJL
Îʈ˜°
Îʈ˜°
{ʈ˜°
Extra Practice
Skills Practice p. S11
Application Practice p. S31
11. When y = 3, NQ = NM = 3, and
QP = MP = 4. So by the def. of
−− −−
−− −−
, NQ NM, and QP MP.
m∠M = m∠Q = 90°, so ∠M ∠Q by the def. of . Thus MNP
QNP by SAS.
10. Theater The lights shining on a stage appear
to form two congruent right triangles.
−− −−
Given EC DB, use SAS to explain why
ECB DBC.
Show that the triangles are congruent for the given
value of the variable.
11. MNP QNP, y = 3
12. XYZ STU, t = 5
8
12. When t = 5, YZ = 24 , ST = 20,
and SU = 22. So by the def. of
−− −− −− −−
−−
, XY ST, YZ TU, and XZ
−−
SU. Thus XYZ STU by
SSS.
Î
ÊÊ£äÞÊÓÊ
ÊÂ
{
-
Óä
ÓÓ
9
ÞÊÊ£
Ó{
<
xÌÊÊ£
/
+
−− −− −−
13. Given: B is the midpoint of DC. AB ⊥ DC
1
{Ì
*
ÞÊÓÊÊÈ
ÌÊÓÊÊÎ
Prove: ABD ABC
Statements
−−
1. B is the mdpt. of DC.
?
−−−−
3. c. ?
−−−−
4. ∠ABD and ∠ABC are rt. .
2. b.
5. ∠ABD ∠ABC
?
−−−−
7. ABD ABC
6. f.
246
Proof:
Reasons
a. Given
−− −−
b. DB CB
−− −−
c. AB ⊥ DC
d. Def. of ⊥
e. Rt. ∠ Thm.
−− −−
f. AB AB
g. SAS Steps 2, 5, 6
1. a.
?
−−−−
2. Def. of mdpt.
3. Given
?
−−−−
?
−−−−
6. Reflex. Prop. of 4. d.
5. e.
7. g.
?
−−−−
Chapter 4 Triangle Congruence
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Visual In Exercise 15, students may forget that they
can use the vertical angles
even though they are not marked
congruent. Remind them to look for
common sides, common angles, and
vertical angles in a diagram.
Which postulate, if any, can be used to prove the triangles congruent?
SAS
14.
SAS
15.
16.
17.
ÓÊvÌ
ÎäÂ
ÓÊvÌ
ÎÊvÌ
neither
neither
ÎÊvÌ
18. Explain what additional information, if any, you would
need to prove ABC DEC by each postulate.
a. SSS
b. SAS
18a. To use SSS,
you need
−− to−−know
that AB
−− DE
−−
and CB CE.
b. To use SAS,
you need
−− to−−know
that CB CE.
19. QRS and TUV
20. ABC and DEF
Q (-2, 0), R (1, -2), S (-3, -2)
21. Given: ∠ZVY ∠WYV,
21a. Given
∠ZVW ∠WYZ,
b. Def. of −−− −−
VW YZ
c. m∠WVY = m∠ZYV
Prove: ZVY WYV
d. Def. of Proof:
e. −−
Given −−
f. VY YV
Statements
g. SAS Steps 6, 5, 7
A (2, 3), B (3, -1), C (7, 2)
D (-3, 1), E (1, 2), F (-3, 5)
Answers
6
19. Check students’ graphs.
QS = TV = √
5 . SR = VU = 4.
QR = TU = √
13 . The are by SSS.
7
8
9
<
20. Check students’ graphs.
AB = √
17 , BC = 5, AC = √
26 ;
DE = √
17 , EF = 5, and DF = 4.
The are not .
−−
−−
22a. Measure AB and AC on 1 truss
−−
−−
and measure DE and DF on the
−− −−
−− −−
other. If AB DE and AC DF,
then the trusses are by SAS.
b. 3.5 ft; by the Pyth. Thm.,
BC ≈ 3.5 ft. Since the are
−− −−
congruent, EF BC.
Reasons
1. ∠ZVY ∠WYV, ∠ZVW WYZ
1. a.
2. m∠ZVY = m∠WYV,
m∠ZVW = m∠WYZ
2. b.
3. m∠ZVY + m∠ZVW =
m∠WYV + m∠WYZ
3. Add. Prop. of =
?
−−−−
?
−−−−
4. ∠ Add. Post.
4. c. ?
−−−−
5. ∠WVY ∠ZYV
−−− −−
6. VW YZ
?
−−−−
?
−−−−
7. Reflex. Prop. of 5. d.
6. e.
7. f. ?
−−−−
8. ZVY WYV
8. g.
?
−−−−
22. This problem will prepare you for the Multi-Step Test Prep
on page 280. The diagram shows two triangular trusses
that were built for the roof of a doghouse.
a. You can use a protractor to check that ∠A and ∠D
are right angles. Explain how you could make just
two additional measurements on each truss to
ensure that the trusses are congruent.
b. You verify that the trusses are congruent and find
−−
that AB = AC = 2.5 ft. Find the length of EF to the
nearest tenth. Explain.
4-4 Triangle Congruence: SSS and SAS
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ge07se_c04_0242_0249.indd 247
Exercise 23 involves
proving triangles
congruent. This
exercise prepares students for the
Multi-Step Test Prep on page 280.
Multi-Step Graph each triangle. Then use the Distance Formula and the
SSS Postulate to determine whether the triangles are congruent.
T (5, 1), U (3, -2), V (3, 2)
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Algebra For Exercises 19
and 20, review the
Distance Formula.
d = √(
x 2 - x 1) 2 + (y 2 - y 1) 2
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Lesson 4-4
247
Construction
To help students construct
the triangle in Exercise 27,
−−
have them construct AB congruent
to one of the segments. Then set
their compasses to the length of
another segment. Have them make
an arc centered at A. Next set their
compasses to the length of the third
segment and draw an arc centered
at B. They should label the intersection of the arcs C. Finally have them
draw ABC.
23. Critical Thinking Draw two isosceles triangles that are
not congruent but that have a perimeter of 15 cm each.
ÈÝÊÊ££
24. ABC ADC for what value of x? Explain why
the SSS Postulate can be used to prove the two
triangles congruent.
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25. Ecology A wing deflector is a triangular structure made of logs that is filled
with large rocks and placed in a stream to guide the current or prevent erosion.
Wing deflectors are often used in pairs. Suppose an engineer wants to build two
wing deflectors. The logs that form the sides of each wing deflector are perpendicular.
How can the engineer make sure that the two wing deflectors are congruent?
Wing deflectors are
designed to reduce the
width-to-depth ratio
of a stream. Reducing
the width increases the
velocity of the stream.
7ˆ˜}Ê`iviV̜ÀÃ
26. Write About It If you use the same two sides and included angle to repeat the
construction of a triangle, are your two constructed triangles congruent? Explain.
If students chose H
for Exercise 29, they
included the length
−−
of AC in the perimeter.
27. Construction Use three segments (SSS) to construct a scalene triangle. Suppose
you then use the same segments in a different order to construct a second triangle.
Will the result be the same? Explain.
Answers
23.
28. Which of the three triangles below can be proven congruent by SSS or SAS?
*
**
***
I and II
−−
24. x = 5.5; by the def. of , AB
−− −− −− −−
−−
BD, and BC DC. AC AC
by the Reflex. Prop. of . Thus
ABC ADC by SSS.
II and III
I and III
29. What is the perimeter of polygon ABCD?
29.9 cm
49.8 cm
39.8 cm
59.8 cm
30. Jacob wants to prove that FGH JKL using SAS.
−− −−
−− −−
He knows that FG JK and FH JL. What additional
piece of information does he need?
∠F ∠J
∠H ∠L
∠G ∠K
∠F ∠G
31. What must the value of x be in order to prove that
EFG EHG by SSS?
1.5
4.67
27. Check students’ constructions;
4.25
yes; if each side is to the corr.
side of the second , they can
be in any order.
−−
32. 1. Draw DB. (Through any 2 pts.
248
Chapter 4 Triangle Congruence
there is exactly 1 line.)
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2. ∠ ADC and ∠BCD are supp.
4-4 READING STRATEGIES
{‡{ #OMPAREAND#ONTRAST
(Given)
−−
−−
3. AD CB (conv. of Same-Side
Int. Thm.)
ge07se_c04_0242_0249.indd 248
4. ∠ADB ∠CBD (Alt. Int. Thm.)
−− −−
5. AD CB (Given)
−− −−
6. DB BD (Reflex. Prop. of )
7. ADB CBD (SAS Steps 5,
4, 6)
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26. Yes; if the have the same
2 side lengths and the same
included ∠ measure, the are by SAS.
248
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25. Measure the lengths of the logs.
If the lengths of the logs in 1
wing deflector match the lengths
of the logs in the other wing
deflector, the will be by SAS
or SSS.
33. 1. ∠QPS ∠TPR (Given)
2. ∠RPS ∠RPS (Reflex. Prop.
of )
3. ∠QPR ∠TPS (Subtr. Prop. of
)
−− −− −− −−
4. PQ PT, PR PS (Given)
5. PQR PTS (SAS Steps 3,
4)
I, II, and III
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CHALLENGE AND EXTEND
32. Given:. ∠ADC and ∠BCD are
−− −−
supplementary. AD CB
Prove: ADB CBD
(Hint: Draw an auxiliary line.)
−− −− −− −−
33. Given: ∠QPS ∠TPR, PQ PT, PR PS
Journal
*
Prove: PQR PTS
+
,
/
-
Algebra Use the following information for Exercises 34 and 35.
Find the value of x. Then use SSS or SAS to write a paragraph
x = 16; KJ−−= HJ−− proof showing that two of the triangles are congruent.
= 72, so KJ HJ
−−
34. m∠FKJ = 2x°
35. FJ bisects ∠KFH.
by def. of .
m∠KFJ = (2x + 6)°
m∠KFJ = (3x + 10)°
x
=
27;
FK
=
FH
=
171,
∠FJK ∠FJH by
−− −−
KJ = 4x + 8
m∠HFJ = (3x - 21)°
so FK FH by the def of .
the
−− Rt.−−∠ Thm.
HJ = 6(x - 4)
FK = 8x - 45
∠KFJ ∠HFJ
the def. of
FJ FJ by the Reflex.
−− by −−
FH = 6x + 9
∠ bisector. FJ FJ by the
Reflex. Prop. of . So
FJK FJH by SAS.
Prop. of . So
FJK FJH by SAS.
SPIRAL REVIEW
Solve and graph each inequality. (Previous course)
x - 8 ≤ 5 x ≤ 26
36. _
37. 2a + 4 > 3a a < 4
2
38. -6m - 1 ≤ -13 m ≥ 2
Have students describe how they
can use color coding to help them
recognize the postulates SSS and
SAS. Have them support their
descriptions with colored sketches.
Have students work in pairs. One
student should draw the diagrams
and write the Given and Prove statements. The second student should
write the proof. Then have students
reverse roles. One proof should be
an example of SSS and the other SAS.
Solve each equation. Write a justification for each step. (Lesson 2-5)
a + 5 = -8
39. 4x - 7 = 21
40. _
41. 6r = 4r + 10
4
Given: EFG GHE. Find each value. (Lesson 4-3)
42. x 86
4-4
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1. Show that ABC DBC,
when x = 6.
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43. m∠FEG 34°
44. m∠FGH 70°
ÎÈ ­ÝÊÊÓ{®Â
ÎÝÊÊ£ä
Use geometry software to complete the following. 1. Check students’ drawings.
1. Draw a triangle and label the vertices A, B, and C.
Draw a point and label it D. Mark a vector from A to B
and translate D by the marked vector. Label the image E.
Draw DE.
Mark ∠BAC and rotate DE
about D by the
marked angle. Mark ∠ABC and rotate DE
about E by
the marked angle. Label the intersection F.
3. Write a conjecture about ABC and DEF. ABC DEF
2. They stay the same size and shape.
4-4 Triangle Congruence: SSS and SAS
4-4 PROBLEM SOLVING
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249
−−−
−−
4. Given: PN bisects MO.
PN ⊥ MO
Prove: MNP ONP
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39–41. See p. A15.
.
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Answers
5SETHEDIAGRAMFOR%XERCISESAND
2.
Check students’
4. Test your conjecture by measuring the sides and angles of ABC and DEF. measurements.
*ÀœLi“Ê-œÛˆ˜}
4RIANGLE#ONGRUENCE333AND3!3
Which postulate, if any, can
be used to prove the triangles
congruent?
2. Drag A, B, and C to different locations.
What do you notice about the two triangles?
{‡{
{ÝÊÊ{
−− −−
−−
∠ ABC ∠DBC, BC BC, and AB
−−
DB. So ABC DBC by
SAS.
Using Technology
,%33/.
Â
ÓÈÂ
"
−−−
−−
1. PN bisects MO. (Given)
−− −−
2. MN ON (Def. of bisect)
−− −−
3. PN PN (Reflex. Prop. of )
−−−
−−
4. PN ⊥ MO (Given)
5. ∠PNM and ∠PNO are rt. .
(Def. of ⊥)
6. ∠PNM ∠PNO (Rt. ∠ Thm.)
7. MNP ONP (SAS Steps 2,
6, 3)
Also available on transparency
Lesson 4-4
249
4-5
Organizer
Predict Other Triangle
Congruence Relationships
Use with Lesson 4-5
Pacing:
Traditional 1 day
1
Block __
day
2
Objective: Use geometry
software to explore triangle
congruence relationships.
Use with Lesson 4-5
Activity 1
GI
Materials: geometry software
<D
@<I
Geometry software can help you investigate whether certain
combinations of triangle parts will make only one triangle.
If a combination makes only one triangle, then this arrangement
can be used to prove two triangles congruent.
1 Construct ∠CAB measuring 45° and
∠EDF measuring 110°.
Online Edition
2 Move ∠EDF so that DE
.
overlays BA
intersect, label the
Where DF
and AC
point G. Measure ∠DGA.
Resources
Technology Lab Activities
4-5 Lab Recording Sheet
Teach
Discuss
Discuss with students that ASA
makes only one triangle but that
SSA does not. Investigate and show
students that SSA makes only one
triangle for the special case when
there is a right angle. Also explain
that AAA does not make only one
triangle.
Check students’ constructions.
Check students’ constructions.
3 Move ∠CAB to the left and right without changing the measures of the angles.
Observe what happens to the size of ∠DGA. It stays the same.
4 Measure the distance from A to D. Try to change the shape of the triangle
without changing AD and the measures of ∠A and ∠D. Check students’ work.
Alternative Approach
Have students work in small groups.
Each person in the group should
draw a different triangle and label it
−−
ABC. Have students construct DF
−−
AC . Students should construct an
angle congruent to ∠ A at ∠D and
construct an angle congruent to ∠C
at F, and label their intersection E.
Then have them cut out ABC and
DEF and place ABC over DEF.
Have students compare the results
and make a conjecture.
Try This
Yes; the stays the same
shape if you do not change AD
or the measures of ∠A and ∠D.
1. Repeat Activity 1 using angle measures of your choice. Are your results the same?
Explain.
2. Do the results change if one of the given angles measures 90°? no
3. What theorem proves that the measure of ∠DGA in Step 2 will always be the same? Third Thm.
4. In Step 3 of the activity, the angle measures in ADG stayed the same as the size
of the triangle changed. Does Angle-Angle-Angle, like Side-Side-Side, make only
one triangle? Explain. No; the ∠ measures may stay the same, but the side lengths can vary.
−−
−−
5. Repeat Step 4 of the activity but measure the length of AG instead of AD. Are your
results the same? Does this lead to a new congruence postulate or theorem?
Check students’ constructions; yes; yes; AAS.
6. If you are given two angles of a triangle, what additional piece of information
is needed so that only one triangle is made? Make a conjecture based on your
findings in Step 5. You need the length of 1 side of the . In an AAS combination,
if 2 corr. and sides are , then only 1 is made.
250
Chapter 4 Triangle Congruence
ge07se_c04_0250_0251.indd 250
KEYWORD: MG7 Resources
250
Chapter 4
11/7/05 2:03:29 PM
Close
Activity 2
Key Concept
−−
1 Construct YZ with a length of 6.5 cm.
Triangles can be proved congruent by ASA, but not by SSA or
AAA. These last two methods do
not always form a unique triangle.
Remind students that they need to
find only one counterexample. That
is, students should find one triangle
with the same SSA or AAA that is not
congruent to the original triangle.
Assessment
−−
2 Using YZ as a side, construct ∠XYZ
measuring 43°.
Journal Have students compare and
contrast ASA, AAA, and SSA. Have
them show examples, including
where SSA forms only one triangle.
3 Draw a circle at Z with a radius of 5 cm.
−−−
Construct ZW, a radius of circle Z.
4 Move W around circle Z. Observe the possible
shapes of YZW.
Try This
7. In Step 4 of the activity, how many different triangles were possible?
Does Side-Side-Angle make only one triangle? many; no
8. Repeat Activity 2 using an angle measure of 90° in Step 2 and a circle with a
radius of 7 cm in Step 3. How many different triangles are possible in Step 4? 1
9. Repeat the activity again using a measure of 90° in Step 2 and a circle with a radius
of 8.25 cm in Step 3. Classify the resulting triangle by its angle measures. rt.
10. Based on your results, complete the following conjecture. In a Side-Side-Angle
combination, if the corresponding nonincluded angles are ? , then only one
−−−−
triangle is possible. rt. 4- 5 Technology Lab
ge07se_c04_0250_0251.indd 251
251
5/8/06 12:50:13 PM
4-5 Technology Lab
251
4-5
Organizer
4-5
Pacing: Traditional 2 days
Triangle Congruence:
ASA, AAS, and HL
Block 1 day
Objectives: Apply ASA, AAS, and
HL to construct triangles and solve
problems.
GI
Prove triangles congruent by using
ASA, AAS, and HL.
<D
@<I
Online Edition
Prove triangles congruent
by using ASA, AAS,
and HL.
Vocabulary
included side
Tutorial Videos
Why use this?
Bearings are used to convey direction,
helping people find their way to
specific locations.
Objectives
Apply ASA, AAS, and HL
to construct triangles and
to solve problems.
Countdown to
Testing Week 8
Participants in an orienteering race use a
map and a compass to find their way to
checkpoints along an unfamiliar course.
Directions are given by bearings, which are
based on compass headings. For example,
to travel along the bearing S 43° E, you face
south and then turn 43° to the east.
An included side is the common side of two
consecutive angles in a polygon. The following
postulate uses the idea of an included side.
+
−−
PQ is the included side
of ∠P and ∠Q.
*
Warm Up
,
−−
−−
1. What are sides AC and BC
−−
called? side AB?
legs; hypotenuse
Postulate 4-5-1
POSTULATE
Angle-Side-Angle (ASA) Congruence
HYPOTHESIS
If two angles and the included
side of one triangle are
congruent to two angles and
the included side of another
triangle, then the triangles
are congruent.
2. Which side is between ∠ A
−−
and ∠C? AC
3. Given DEF and GHI, if ∠D
∠G and ∠E ∠H, why is
∠F ∠I? Third Thm.
EXAMPLE
1
Also available on transparency
ABC DEF
Problem-Solving Application
Organizers of an orienteering race are
planning a course with checkpoints A,
B, and C. Does the table give enough
information to determine the location
of the checkpoints?
1
Euclid thought he had proved ASA.
It has since been postulated, and
AAS and HL have been presented as
theorems.
CONCLUSION
Bearing
Distance
A to B
N 55° E
7.6 km
B to C
N 26° W
C to A
S 20° W
Understand the Problem
The answer is whether the information in the table can
be used to find the position of checkpoints A, B, and C.
List the important information: The bearing from A
to B is N 55° E. From B to C is N 26° W, and from C
to A is S 20° W. The distance from A to B is 7.6 km.
252
Chapter 4 Triangle Congruence
1 Introduce
ge07se_c04_0252_0259.indd 252
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Motivate
Give students a map of the area that contains
their school. Have them plot a triangle, using the
school, their house, and another location as vertices. Ask them to calculate the measures of the
angles of the triangle. Explain to students that
angles are important in navigation where directions are given by bearings.
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252
Chapter 4
%XPLAIN HOWYOUCOULDUSETHEABOVE
CONJECTURETOSHOWTHATN+,.N-.,
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+
Explorations and answers are provided in the
Explorations binder.
8/16/05 3:24:20 PM
Inclusion Explain that the
included side is named
by the two vertices of the
2 Make a Plan
Draw the course using vertical lines to show north-south
directions. Then use these parallel lines and the alternate
interior angles to help find angle measures of ABC.
angles.
ÓÈÂ
ÓäÂ
ÓäÂ
3 Solve
xxÂ
Îx ǰÈʎ“
m∠CAB = 55° - 20° = 35°
m∠CBA = 180° - (26° + 55°) = 99°
You know the measures of ∠CAB and ∠CBA and the length of the included
−−
side AB. Therefore by ASA, a unique triangle ABC is determined.
4 Look Back
One and only one triangle can be made using the information in the table,
so the table does give enough information to determine the location of all
the checkpoints.
1. What if...? If 7.6 km is the distance from B to C, is there
enough information to determine the location of all the
checkpoints? Explain.
Yes; the is uniquely determined by AAS.
2
EXAMPLE
Applying ASA Congruence
Determine if you can use ASA to prove UVX WVX. Explain.
8
∠UXV ∠WXV as given. Since ∠WVX is
a right angle that forms a linear pair with
−− −−
∠UVX, ∠WVX ∠UVX. Also VX VX
by the Reflexive Property. Therefore
UVX WVX by ASA.
1
Bearing
Distance
A to B
N 65° E
8 mi
B to C
N 24° W
C to A
S 20° W
Example 2
Determine if you can use ASA to
prove the triangles congruent.
Explain.
no; no included side
−− −−
By the Alt. Int. Thm., ∠KLN ∠MNL. NL LN by
the Reflex. Prop. No other congruence relationships
can be determined, so ASA cannot be applied.
Construction
Example 1
A mailman has to collect mail
from mailboxes at A and B and
drop it off at the post office at C.
Does the table give enough information to determine the location
of the mailboxes and the post
office? yes
7
6
2. Determine if you can use ASA to
prove NKL LMN. Explain.
Additional Examples
Also available on transparency
Congruent Triangles Using ASA
INTERVENTION
Questioning Strategies
Use a straightedge to draw a segment and two angles,
or copy the given segment and angles.
E X A M P LE
−−
Construct CD congruent to
the given segment.
E X A M P LE
CDE
Construct ∠C congruent
to one of the angles.
Construct ∠D congruent
to the other angle.
253
2 Teach
ge07se_c04_0252_0259.indd 253
5/8/06
Guided Instruction
Review with students how to read bearings before presenting examples. If more
explanation is needed for the drawing in
Example 1, give students the following
directions:
1. Draw A.
2. Find B.
 with
3. Use a straightedge and draw AC
the bearing from A being N 20° E.
Through Auditory Cues
2
• What transformation take place to
change position from one triangle
to the orientation of the second
triangle?
Label the intersection of
the rays as E.
4-5 Triangle Congruence: ASA, AAS, and HL
1
• How are bearings used when
finding the distance between two
points?
Reading Math Mark the
given information on the
diagram before starting a
proof. Put an S or an A next to each
step
12:51:27
PM in the proof to
ENGLISH
LANGUAGE
indicate which side
LEARNERS
or angle is given.
As students work through the lesson have
them identify orally whether the triangles
are congruent by HL, ASA, AAS, SAS, or
SSS. Then have them work with a partner
to identify the congruent parts and to
determine if the congruent pairs of angles
or sides are corresponding parts.
With students, practice identifying ASA,
AAS, and HL from diagrams before writing
proofs.
Lesson 4-5
253
You can use the Third Angles Theorem to prove another congruence relationship
based on ASA. This theorem is Angle-Angle-Side (AAS).
Additional Examples
Theorem 4-5-2
Example 3
Angle-Angle-Side (AAS) Congruence
THEOREM
Use AAS to prove the triangles
congruent.
HYPOTHESIS
If two angles and a nonincluded
side of one triangle are
congruent to the corresponding
angles and nonincluded side
of another triangle, then the
triangles are congruent.
Given: ∠X ∠V, ∠YZW ∠YWZ,
−− −−
XY VY
Prove: XYZ VYW
8
<
GHJ KLM
9
7
6
Angle-Angle-Side Congruence
PROOF
ȜVWY ISSUPP
TOȜYWZ.
$EFOFSUPPѓ
ȜYZW Ɂ ȜYWZ
'IVEN
ȜYZX Ɂ ȜYWV
ȜX Ɂ ȜV
Ɂ3UPPS4HM
'IVEN
̱XYZ Ɂ ̱VYW
Statements
$EFOFSUPPѓ
−− −−−
Given: ∠G ∠K, ∠J ∠M, HJ LM
Prove: GHJ KLM
Proof:
ȜXZY ISSUPP
TOȜYZW.
CONCLUSION
Reasons
1. ∠G ∠K, ∠J ∠M
1. Given
2. ∠H ∠L
−− −−−
3. HJ LM
2. Third Thm.
4. GHJ KLM
4. ASA Steps 1, 3, and 2
3. Given
!!3
XY Ɂ VY
'IVEN
Also available on transparency
EXAMPLE
3
Using AAS to Prove Triangles Congruent
Use AAS to prove the triangles congruent.
−− −− −− −−
Given: AB ED, BC DC
Prove: ABC EDC
Proof:
INTERVENTION
Questioning Strategies
ÊɁÊ
E X AM P LE
3
ˆÛi˜
• If AAS is used as the method of
proof, can the triangles also be
proved congruent using ASA?
ÊȡÊ
ˆÛi˜
̱
ÊɁÊ̱
-
ȜÊɁÊȜ
Ì°Ê˜Ì°ÊѐÊ/…“°
Answers to Check It Out!
3.
ȜÊɁÊȜ
Ì°Ê˜Ì°ÊѐÊ/…“°
3. Use AAS to prove the triangles congruent.
−−
Given: JL bisects ∠KLM. ∠K ∠M
Prove: JKL JML
JL bisects ȜKLM.
'IVEN
ȜKLJ Ɂ ȜMLJ
$EFOFȜBISECTOR
ȜK Ɂ ȜM
JL Ɂ JL
'IVEN
2EFLEX
0ROPOFɁ
̱JKL Ɂ ̱JML
There are four theorems for right triangles that are not used for acute or obtuse
triangles. They are Leg-Leg (LL), Hypotenuse-Angle (HA), Leg-Angle (LA), and
Hypotenuse-Leg (HL). You will prove LL, HA, and LA in Exercises 21, 23, and 33.
254
Chapter 4 Triangle Congruence
!!3
ge07se_c04_0252_0259.indd 254
254
Chapter 4
8/16/05 3:24:27 PM
Theorem 4-5-3
Visual For Example 4, it
may be less confusing if
students redraw the triangles separately and then carefully
label the triangles and mark the congruent corresponding parts.
Hypotenuse-Leg (HL) Congruence
THEOREM
HYPOTHESIS
If the hypotenuse and a leg of
a right triangle are congruent
to the hypotenuse and a leg of
another right triangle, then the
triangles are congruent.
CONCLUSION
ABC DEF
You will prove the Hypotenuse-Leg Theorem in Lesson 4-8.
EXAMPLE
4
Applying HL Congruence
Determine if you can use the HL Congruence Theorem to prove the
triangles congruent. If not, tell what else you need to know.
A VWX and YXW
6
According to the diagram, VWX and
YXW are right triangles that share
−−− −−− −−−
hypotenuse WX. WX XW by the Reflexive
−−− −−
Property. It is given that WV XY,
therefore VWX YXW by HL.
<
Inclusion For Example 4
show that AAS could be
used as the method of
proof. Include an illustration showing
why SSA cannot be used to prove
two triangles congruent. If the triangles are right triangles, then you
would use HL.
DAB CAB
9
7
8
B VWZ and YXZ
This conclusion cannot be proved by HL. According
to the diagram, VWZ and YXZ are right triangles,
−−− −−
−−−
and WV XY. You do not know that hypotenuse WZ
−−
is congruent to hypotenuse XZ.
4. Yes; it is given
−− −− −− −−
that AC DB. BC CB
by the Reflex. Prop.
of . Since ∠ABC and
∠DCB are rt. , ABC
and DCB are rt. .
ABC DCB by HL.
Additional Examples
Example 4
4. Determine if you can use the HL
Congruence Theorem to prove
ABC DCB. If not, tell what
else you need to know.
Determine if you can use the
HL Congruence Theorem to
prove the triangles congruent.
If not, tell what else you need
to know.
A.
THINK AND DISCUSS
yes
1. Could you use AAS to prove that these
two triangles are congruent? Explain.
ÈäÂ
{xÂ
Èä {xÂ
2. The arrangement of the letters in ASA matches the arrangement
of what parts of congruent triangles? Include a sketch to support
your answer.
B.
No; you need the hyp. .
3. GET ORGANIZED Copy and complete the graphic organizer.
In each column, write a description of the method and then
sketch two triangles, marking the appropriate congruent parts.
Also available on transparency
*ÀœÛˆ˜}Ê/Àˆ>˜}iÃÊ
œ˜}ÀÕi˜Ì
iv°Êœv̱ÊɁ
---
--
-
-
INTERVENTION
Questioning Strategies
7œÀ`Ã
*ˆVÌÕÀiÃ
E X A M P LE
4-5 Triangle Congruence: ASA, AAS, and HL
Summarize
Review how to identify when you should
use AAS, HL, or ASA to prove triangles
congruent. Remind students that three
pairs of congruent corresponding parts are
necessary for AAS and ASA. For HL they
first must prove that the triangles are right
triangles.
• What type of triangle must be given
to use HL as a method of proof?
Answers to Think and Discuss
3 Close
ge07se_c04_0252_0259.indd 255
255
4
1. No; the sides are not corr. sides.
and INTERVENTION
Diagnose Before the Lesson
4-5 Warm Up, TE p. 252
2. Possible answer: corr. and sides
5/8/06 12:53:53 PM
3. See p. A4.
Monitor During the Lesson
Check It Out! Exercises, SE pp. 253–255
Questioning Strategies, TE pp. 253–255
Assess After the Lesson
4-5 Lesson Quiz, TE p. 259
Alternative Assessment, TE p. 259
Lesson 4-5
255
4-5
4-5 Exercises
Exercises
KEYWORD: MG7 4-5
KEYWORD: MG7 Parent
GUIDED PRACTICE
Assignment Guide
−−
1. Vocabulary A triangle contains ∠ABC and ∠ACB with BC “closed in” between
them. How would this help you remember the definition of included side?
Assign Guided Practice exercises
as necessary.
SEE EXAMPLE
If you finished Examples 1–2
Basic 9–12, 17, 25
Average 9–12, 17, 21, 25
Advanced 9–12, 17, 21, 25, 32
Surveying Use the table for Exercises 2 and 3.
A landscape designer surveyed the boundaries of
a triangular park. She made the following table for
the dimensions of the land.
1
p. 252
A to B
Bearing
If you finished Examples 1–4
Basic 9–17, 19, 20, 22, 25,
26–30, 35–39
Average 9–20, 22, 24–30, 34,
35–39
Advanced 9–16, 18, 19, 21–39
E
Distance
115 ft
B to C
C to A
S 25° E
N 62° W
?
SEE EXAMPLE
Yes, the is determined by AAS.
4. VRS and VTS, given that
−−
VS bisects ∠RST and ∠RVT
p. 253
5. DEH and FGH
-
−−
1. The included side BC is enclosed
between ∠ ABC and ∠ ACB.
ƒ
+
6. Use AAS to prove the triangles congruent.
3
Given: ∠R and ∠P are right angles.
−− −−
QR SP
Prove: QPS SRQ
p. 254
B
*
Proof:
ƒ
>°ÊÊÊʶÊÊÊ
C
,iviÝ°Ê*Àœ«°ÊœvÊɁ
4. Yes; by the def. of bisector, ∠TSV
−−
∠RSV, and ∠TVS ∠RVS. SV
−−
SV by the Reflex. Prop. of .
So VRS VTS by ASA.
7. Yes; it is given that ∠D and ∠B
−− −−
are rt. and AD BC. ABC
and CDA are rt. by def.
−− −−
AC CA by the Reflex. Prop. of
. So ABC CDA by HL.
+,ÊȡÊ*-
L°ÊÊÊʶÊÊÊ
̱+*-ÊɁÊ̱-,+
ˆÛi˜
Ì°Ê˜Ì°ÊѐÊ/…“°
`°ÊÊÊʶÊÊÊ
Ȝ,Ê>˜`ÊȜ*Ê>ÀiÊÀÌ°Êѐ°
Ȝ,ÊɁÊȜ*
ˆÛi˜
V°ÊÊÊʶÊÊÊ
p. 255
7. ABC and CDA
256
−− −− a.QS SQ
b. ∠RQS ∠PSQ
c. Rt ∠ Thm.
d. AAS
No; you need to
know
−− that
−−
VX VZ.
6
9
<
Chapter 4 Triangle Congruence
LESSON
4-5
Practice A
4-5 PRACTICE A
Triangle Congruence: ASA, AAS, and HL
1. X and Z
XZ
3. Y and Z
YZ
9
_
YX
2. Y and X
_
8
:
ge07se_c04_0252_0259.indd Write
256ASA (Angle-Side-Angle Congruence), AAS (Angle-Angle-Side Congruence),
5/8/06 12:54:02 PM
or HL (Hypotenuse-Leg Congruence) next to the correct postulate.
4. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse
and a leg of another right triangle, then the triangles are congruent.
HL
5. If two angles and a nonincluded side of one triangle are congruent to the
corresponding angles and nonincluded side of another triangle, then the
triangles are congruent.
AAS
6. If two angles and the included side of one triangle are congruent to two angles
and the included side of another triangle, then the triangles are congruent.
ASA
For Exercises 7–9, tell whether you can use each congruence
theorem to prove that OABC ODEF. If not, tell what else
you need to know.
7. Hypotenuse-Leg
_
!
$
"
#
&
%
_
No; you need to know that AC DF .
8. Angle-Side-Angle
Yes, if you use Third
Thm. first.
9. Angle-Angle-Side
Yes
10. A standard letter-sized envelope is a 9 _1_ -in.2
by-4-in. rectangle. The envelope is folded
and glued from a sheet of paper shaped
like the figure. Use the phrases in the
word bank to complete this proof.
9 1–2 in.
Prove: NIJK NLMN
Statements
-
.
,
Given,
ASA,
Definition of rectangle
Reasons
1. IJK LMN, IKJ LNM
1. a.
Given
2. JK MN
2. b.
Definition of rectangle
3. NIJK NLMN
3. c.
ASA
_
*
+
)
4 in.
Given: JMNK is a rectangle. IJK LMN, IKJ LNM
_
Chapter 4
,
8. XYV and ZYV
Name the included side for each pair of
consecutive angles._
KEYWORD: MG7 Resources
No; you need to
know that a pair of
corr. sides are .
Determine if you can use the HL Congruence Theorem to prove the triangles
congruent. If not, tell what else you need to know.
SEE EXAMPLE 4
8
256
,
SEE EXAMPLE
6
Answers
ƒ
B
C
Determine if you can use ASA to prove the triangles congruent. Explain.
2
/
A
?
3. Does the table have enough information to
determine the locations of points A, B, and C? Explain.
Quickly check key concepts.
Exercises: 10, 12, 13, 14, 16, 22
FT
115 ft
2. Draw the plot of land described by the table.
Label the measures of the angles in the triangle.
Homework Quick Check
2.
A
PRACTICE AND PROBLEM SOLVING
9–10
11–12
13
14–15
""
Surveying Use the table for Exercises 9 and 10.
From two different observation towers a fire is sighted. The locations of the towers
are given in the following table.
Independent Practice
For
See
Exercises Example
1
2
3
4
X to Y
Bearing
E
Distance
6 km
X to F
Y to F
N 53° E
N 16° W
?
For Exercise 9, students may label
the wrong angles 53° and 16°.
Review that N 53° E means 53° east
of north, instead of 53° north of
east.
?
Extra Practice
Skills Practice p. S11
9. Draw the diagram formed by observation tower X, observation tower Y,
and the fire F. Label the measures of the angles.
Application Practice p. S31
Transformations For
Exercises 16 and 17, it
helps students increase
their visual discrimination if they
can rotate, reflect, and slide figures
mentally to determine what transformations would make the triangles’
orientations identical.
10. Is there enough information given in the table to pinpoint the location of
the fire? Explain. Yes; the is uniquely determined by ASA.
Determine if you can use ASA to prove the triangles congruent. Explain.
Math History
11. MKJ and MKL
−− −−
13. Given: AB DE, ∠C ∠F
Prove: ABC DEF
12. RST and TUR
No; you
need to
,
know that
∠MKJ ∠MKL.
1
/
Euclid wrote the
mathematical text
The Elements around
2300 years ago. It may
be the second most
reprinted book in history.
Proof:
ȜÊ>˜`ÊȜÊ>ÀiÊÀÌ°Êѐ°
>°ÊÊÊʶÊÊÊ
ˆÛi˜
,Ì°ÊȜÊɁÊ/…“°
Ê,,",
,/
-
Answers
9.
a. ∠A ∠D
b. Given
c. ∠C ∠F
d. AAS
F
ƒ
X
ƒ
KM
ƒ
Y
Determine if you can use the HL Congruence Theorem to prove the triangles
congruent. If not, tell what else you need to know.
12. Yes; by the Alt. Int. Thm. ∠SRT
−−
∠UTR, and ∠STR ∠URT. RT
−−
TR by the Reflex. Prop. of .
So RST TUR by ASA.
−−
15. Yes; E is a mdpt. So by def., BE
−−
−− −−
CE, and AE DE. ∠ A and ∠D
are by the Rt. ∠ Thm. By
def., ABE and DCE are rt. .
So ABE DCE by HL.
14. GHJ and JKG
18.
ÊɁÊ
̱
ÊɁÊ̱
L°ÊÊÊʶÊÊÊ
`°ÊÊÊʶÊÊÊ
V°ÊÊÊʶÊÊÊ
ˆÛi˜
No; you need to
know that ∠K and
∠H are rt. .
15. ABE and DCE,
given that E is
the midpoint
−−
−−
of AD and BC
Multi-Step For each pair of triangles write a triangle congruence statement.
Identify the transformation that moves one triangle to the position of the
other triangle.
16.
17. ADB CDB; reflection
,
-
+
FEG QSR; rotation
18. Critical Thinking Side-Side-Angle (SSA) cannot be used to prove two
triangles congruent. Draw a diagram that shows why this is true.
4-5 Triangle Congruence: ASA, AAS, and HL
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Lesson 4-5
257
Exercise 19 involves
proving that the
triangles that form
a truss are congruent. This exercise
prepares students for the Multi-Step
Test Prep on page 280.
Construction For Exercise
25, have students draw a
5 cm segment and a 10 cm
−−
segment. Have them construct AB
congruent to the 5 cm segment and
−−

AX perpendicular to AB at A. Then
have them set their compasses to
the length of the 10 cm segment
and draw an arc centered at A. Have
them label the intersection of the
arc and 
AX as C. Finally, have them
draw right ABC.
19. This problem will prepare you for the Multi-Step Test Prep on page 280.
A carpenter built a truss to support the roof of a doghouse.
−− −−
a. The carpenter knows that KJ MJ. Can the carpenter
conclude that KJL MJL? Why or why not?
b. Suppose the carpenter also knows that ∠JLK is
a right angle. Which theorem can be used to
show that KJL MJL? HL
20.
ÌʈÃÊ}ˆÛi˜Ê̅>ÌÊÊȡÊ°ÊÞÊ̅i
Ì°Ê˜Ì°ÊѓÊ/…“°]ÊȜÊɁÊȜ°
ȜÊɁÊȜÊLÞÊ̅iÊ,Ì°ÊȜÊɁÊ/…“°ÊÞÊ
̅iÊ,iviÝ°Ê*Àœ«°ÊœvÊɁ]ÊÊɁÊ°
-œÊLÞÊ-]Ê̱ÊɁÊ̱°
22. Use AAS to prove the triangles congruent.
−− −− −− −−
Given: AD BC, AD CB
Prove: AED CEB
Statements
−−− −−
1. AD BC
19a. No; there is not enough information given to use any of the
congruence theorems.
3. c.
?
−−−−
2. b. ?
−−−−
3. Vert. Thm.
4. d.
3. Given
2. ∠DAE ∠BCE
?
−−−−
?
−−−−
5. e. ?
−−−−
D
F
Reasons
a. Given
b. Alt. Int. Thm.
c. ∠AED
∠CEB
−− −−
d. AD CB
e. AED CEB
f. AAS Steps 2, 3, 4
1. a.
4. f.
?
−−−−
23. Prove the Hypotenuse-Angle (HA) Theorem.
−−− −− −− −−−
Given: KM ⊥ JL, JM LM, ∠JMK ∠LMK
Prove: JKM LKM
E
It is given that ABC and DEF
−− −− −− −−
are rt. . AC DF, BC EF,
and ∠C and ∠F are rt. . ∠C
∠F by the Rt. ∠ Thm. Thus
ABC DEF by SAS.
−−
−−
23. 1. KM ⊥ JL (Given)
2. ∠ JKM and ∠LKM are rt. .
(Def. of ⊥)
3. ∠ JKM ∠LKM (Rt. ∠ Thm.)
−− −−
4. JM LM, ∠ JMK ∠LMK
(Given)
5. JKM LKM (AAS Steps
3, 4)
24. Since 2 sides and the included ∠
are equal in measure and therefore , you could prove the using SAS. You could also use HL
since the are rt. .
Answers
B
ʈÃÊ̅iʅޫ°ÊœvÊLœÌ…ÊÀÌ°Êє°ÊÊɁÊ
ÊLÞÊ̅iÊ,iviÝ°Ê*Àœ«°ÊœvÊɁ°Ê-ˆ˜ViÊ
̅iʜ««°ÊÈ`iÃʜvÊ>ÊÀiVÌ°Ê>ÀiÊɁ]ÊÊɁÊ
°Ê-œÊLÞÊ]Ê̱ÊɁÊ̱°
Proof:
C
21. Write a paragraph proof of the Leg-Leg (LL) Congruence Theorem. If the legs of
one right triangle are congruent to the corresponding legs of another right triangle,
the triangles are congruent.
21. A
"
20. Proof B is
incorrect. The corr.
sides are not in
the correct order.
8
Two proofs that EFH GHF
are given. Which is incorrect? Explain the error.
/////ERROR ANALYSIS/////
!
24. Write About It The legs of both right DEF and right RST are 3 cm and 4 cm.
They each have a hypotenuse 5 cm in length. Describe two different ways you could
prove that DEF RST.
25. Construction Use the method for constructing perpendicular lines to construct
a right triangle. Check students’ constructions.
8
26. What additional congruence statement is necessary to prove
XWY XVZ by ASA?
−− −−−
∠XVZ ∠XWY
VZ WY
−− −−
∠VUY ∠WUZ
XZ XY
258
7
6
1
9
<
Chapter 4 Triangle Congruence
31. Yes; the sum of the ∠ measures
in each must be 180°, which
makes it possible to solve for x ge07se_c04_0252_0259.indd
and y. The value of x is 15, and
the value of y is 12. Each has measuring 82°, 68°, and
−− −−
30°. VU VU by the Reflex. Prop.
of . So VSU VTU by ASA
or AAS.
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27. Which postulate or theorem justifies the congruence
statement STU VUT?
ASA
HL
SSS
SAS
-
6
/
1
28. Which of the following congruence statements is true?
∠A ∠B
AED CEB
−− −−
CE DE
AED BEC
For Exercise 27, students may assume
that the hypotenuses
−−
are congruent. Since TU is a common side, the triangles are congruent by SAS, not HL.
30. Check students’
drawings and
29. In RST, RT = 6y - 2. In UVW, UW = 2y + 7. ∠R ∠U, and ∠S ∠V.
constructions;
What must be the value of y in order to prove that RST UVW?
since the lengths
1.25
2.25
9.0
11.5
of the corr. sides
of the 2 are not 30. Extended Response Draw a triangle. Construct a second triangle that has the
equal, the 2 are
same angle measures but is not congruent. Compare the lengths of each pair of
not even if the
corresponding sides. Consider the relationship between the lengths of the sides
corr. have the
and the measures of the angles. Explain why Angle-Angle-Angle (AAA) is not a
congruence principle.
same measure.
Journal
Have students describe how proving
two triangles congruent by HL is different from the SAS method.
CHALLENGE AND EXTEND
31. Sports This bicycle frame includes VSU
and VTU, which lie in intersecting planes.
From the given angle measures, can you
conclude that VSU VTU? Explain.
(
6
)
1 °
1x - _
m∠VUT = 5_
2
2
m∠UTV = (4x + 8)°
m∠VUS = (7y - 2)°
2y°
m∠USV = 5_
3
m∠SVU = (3y - 6)°
/
-
m∠TVU = 2x °
1
32. Given: ABC is equilateral. C is the midpoint of
−−
DE. ∠DAC and ∠EBC are congruent
and supplementary.
Prove: DAC EBC
4-5
33. Write a two-column proof of the Leg-Angle (LA) Congruence Theorem. If a leg and
an acute angle of one right triangle are congruent to the corresponding parts of
another right triangle, the triangles are congruent. (Hint: There are two cases
to consider.)
34. Third Thm.;
if the third pair
of are , then
the are also
by AAS.
Group students in pairs and have
each student cut out three large
triangles. Have them then fold each
over to create two congruent triangles. Each student should mark
off given information on the triangles
and then give them to their partner
to decide if the triangles are congruent by AAS, ASA, or HL.
Identify the postulate or
theorem that proves the
triangles congruent.
34. If two triangles are congruent by ASA, what theorem could you use to prove that the
triangles are also congruent by AAS? Explain.
1.
£xäÂ
Identify the x- and y-intercepts. Use them to graph each line. (Previous course)
1 x + 4 8; 4
35. y = 3x - 6 2; -6
36. y = -_
37. y = -5x + 5 1; 5
2
38. Find AB and BC if AC = 10. (Lesson 1-6)
xΰ£Â
AB = 6; BC = 8
ÝÓ
36.9°
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259
4. Given: ∠FAB ∠GED, ∠ACB
−− −−
∠DCE, AC EC
Prove: ABC EDC
Answers
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& N"#!N('!BY(,
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-
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3.
4-5 PROBLEM SOLVING
-ELANIEISATHOLEONAMINIATUREGOLFCOURSE3HE
WALKSEASTMETERSTOHOLE3HETHENFACESSOUTH
TURNS—WESTANDWALKSTOHOLE&ROMHOLESHE
FACESNORTHTURNS—WESTANDWALKSTOHOLE
HL
™ä ™äÂ
Ý ÓÈ
4-5 Triangle Congruence: ASA, AAS, and HL
5SETHEFOLLOWINGINFORMATIONFOR%XERCISESAND
2.
ASA
£xÂ
£xäÂ
SPIRAL REVIEW
39. Find m∠C. (Lesson 4-2)
£xÂ
32. 1. ABC is equil. (Given)
−− −−
2. AC BC (Def. of equil.)
−−
3. C is the mdpt. of DE . (Given)
−− −−
4. DC EC (Def. of mdpt.)
5. ∠DAC and ∠EBC are and supp.
(Given)
6. ∠DAC and ∠EBC are rt. . ( that
are and supp. are rt. .)
7. DAC and EBC are rt. . (Def. of
rt. )
8. DAC EBC (HL Steps 4, 2)
33, 35–37. See pp. A15–A16.
8/16/05 3:24:49 PM
Proof:
1. ∠FAB ∠GED (Given)
2. ∠BAC is a supp. of ∠FAB; ∠DEC
is a supp. of ∠GED. (Def. of
supp. )
3. ∠BAC ∠DEC ( Supp. Thm.)
−− −−
4. ∠ACB ∠DCE; AC EC (Given)
5. ABC EDC (ASA Steps 3, 4)
Also available on transparency
Lesson 4-5
259
4-6
Organizer
Pacing: Traditional __12 day
Block
__1 day
4
Objective: Use CPCTC to prove
GI
@<I
Why learn this?
You can use congruent triangles to
estimate distances.
Objective
Use CPCTC to prove
parts of triangles are
congruent.
parts of triangles are congruent.
<D
Triangle
Congruence: CPCTC
4-6
Online Edition
Tutorial Videos
CPCTC is an abbreviation for the phrase
“Corresponding Parts of Congruent
Triangles are Congruent.” It can be used
as a justification in a proof after you
have proven two triangles congruent.
Vocabulary
CPCTC
Countdown to
Testing Week 8
1
EXAMPLE
Warm Up
To design a bridge across a canyon, you
need to find the distance from A to B.
Locate points C, D, and E as shown in
the figure. If DE = 600 ft, what is AB?
∠D ∠B, because they are both right angles.
−− −−
DC CB ,because DC = CB = 500 ft.
∠DCE ∠BCA, because vertical angles
are congruent. Therefore DCE BCA
−− −−
by ASA or LA. By CPCTC, ED AB, so
AB = ED = 600 ft.
SSS, SAS, ASA,
AAS, and HL use
corresponding parts
to prove triangles
congruent. CPCTC
uses congruent
triangles to prove
corresponding
parts congruent.
1. If ABC DEF, then ∠ A −−
−−
? and BC −−−
? . ∠D, EF
−−−
2. What is the distance between
(3, 4) and (-1, 5)? √
17
3. If ∠1 ∠2, why is a b?
Conv. of Alt. Int. Thm.
Engineering Application
Ó
xääÊvÌ xääÊvÌ
ÎäÊvÌ
£™ÊvÌ
ÎäÊvÌ
L
4. List methods used to prove
two triangles congruent.
SSS, SAS, ASA, AAS, HL
2
EXAMPLE
Also available on transparency
1. A landscape architect sets up the
triangles shown in the figure to
find the distance JK across a pond.
What is JK? 41 ft
>
£
£™ÊvÌ
{£ÊvÌ
Proving Corresponding Parts Congruent
−− −−
Given: AB DC, ∠ABC ∠DCB
Prove: ∠A ∠D
Proof:
ÊɁÊ
ˆÛi˜
Ȝ
ÊɁÊȜ
̱
ÊɁÊ̱
ȜÊɁÊȜ
ˆÛi˜
--
*
/
ÊɁÊ
Q: What do you write as the reason
when using corresponding parts of
congruent triangles in a proof?
,iviÝ°Ê*Àœ«°ÊœvÊɁ
−−
2. Given: PR bisects ∠QPS and ∠QRS.
−− −−
Prove: PQ PS
*
A: See Peas Eat Easy! (CPCTC)
+
,
-
260
Chapter 4 Triangle Congruence
1 Introduce
ge07se_c04_0260_0265.indd 260
E X P L O R AT I O N
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260
Chapter 4
%XPLAIN HOWYOUCOULD
0
PROVETHATN013N231
1
Motivate
Point out that congruent triangles can be used
to find the distance between two points that is
difficult to measure, such as the distance across
a lake. Show designs by M. C. Escher to demonstrate that the size of one part determines the size
of another. Students will learn that you can base
assumptions about parts of a triangles on information about other parts.
Explorations and answers are provided in the
Explorations binder.
5/8/06 1:02:36 PM
EXAMPLE
3
Using CPCTC in a Proof
−− −− −− −−
Given: EG DF, EG DF
−− −−
Prove: ED GF
Proof:
Additional Examples
Statements
−− −−
1. EG DF
−− −−
2. EG DF
Work backward
when planning
a proof. To show
−− −−
that ED GF, look
for a pair of angles
that are congruent.
Then look for
triangles that contain
these angles.
Example 1
Reasons
1. Given
2. Given
3. ∠EGD ∠FDG
−−− −−−
4. GD DG
3. Alt. Int. Thm.
5. EGD FDG
5. SAS Steps 1, 3, and 4
6. ∠EDG ∠FGD
−− −−
7. ED GF
6. CPCTC
£nʓˆ
4. Reflex. Prop. of £xʓˆ
£äʓˆ
A and B are on the edges of a
ravine. What is AB? 18 mi
7. Converse of Alt. Int. Thm.
−−−
−−
3. Given: J is the midpoint of KM and NL.
−− −−−
Prove: KL MN
£xʓˆ
£äʓˆ
Example 2
9
−−
Given: YW bisects
−−
XZ.
−− −− 8
XY YZ.
<
7
Prove: ∠XYW ∠ZYW
You can also use CPCTC when triangles are on a coordinate plane.
You use the Distance Formula to find the lengths of the sides of each triangle.
Then, after showing that the triangles are congruent, you can
make conclusions about their corresponding parts.
YW BISECTS XZ.
'IVEN
XW Ɂ ZW
$EFOFBISECTOR
EXAMPLE
4
Using CPCTC in the Coordinate Plane
Given: A(2, 3), B(5, -1), C(1, 0),
D(-4, -1), E(0, 2), F(-1, -2)
Prove: ∠ABC ∠DEF
{
DE =
= √
9 + 16 = √
25 = 5
BC =
2
EF =
= √16
+ 1 = √
17
AC =
4. RT = JL = √
5,
10 ,
RS = JK = √
and ST = KL = √
17 .
So JKL RST
by SSS. ∠JKL ∠RST
by CPCTC.
ä Ó
#0#4#
Example 3
"
−− −−
Given: NO MP,
∠N ∠P *
−− −−
Prove: MN OP
−− −−
1. ∠N ∠P; NO MP (Given)
(0 - (-4)) + (2 - (-1))
√
2
2
= √
16 + 9 = √
25 = 5
(1 - 5) + (0 - (-1))
√
2
333
ȜXYW Ɂ ȜZYW
{
(5 -2)2 + (-1 - 3)2
√
(-1 - 0)2 + (-2 - 2)2
√
= √1
+ 16 = √
17
(1 - 2)2 + (0 - 3)2
√
DF =
= √1
+ 9 = √
10
(-1 - (-4)) + (-2 - (-1))
√
2
2
Prove: ∠JKL ∠RST
ge07se_c04_0260_0265.indd 261
Summarize
Present CPCTC as a way to use congruent triangles to prove corresponding parts
congruent. Have students complete fill-inthe-blank proofs using CPCTC. Be sure students know the importance of first proving
triangle congruence before using CPCTC to
prove parts congruent. Review the Distance
Formula before Example 4.
Remind students what CPCTC means.
Point out that the middle of this statement
implies that students first prove triangles
congruent and then make conclusions
about the corresponding parts. Remind
students that they can use the Distance
Formula when proving triangle congruence
in the coordinate plane.
Auditory Have students recite
the meaning of CPCTC with
emphasis on the words congruent triangles.
261
3 Close
Guided Instruction
Answers to Check It Out!
4. MNO OPM (AAS)
Example 4
4. Given: J(-1, -2), K(2, -1), L(-2, 0), R(2, 3), S(5, 2), T(1, 1)
4-6 Triangle Congruence: CPCTC
2. ∠NOM ∠PMO (Alt. Int. Thm.)
−−− −−−
3. MO MO (Reflex. Prop of )
5. ∠NMO ∠POM (CPCTC)
−− −−
6. MN OP (Conv. of Alt. Int. Thm.)
= √
9 + 1 = √
10
−− −−
−− −− −− −−
So AB DE, BC EF, and AC DF. Therefore ABC DEF by SSS,
and ∠ABC ∠DEF by CPCTC.
2 Teach
2EFLEX0ROPOFɁ
̱XYW Ɂ ̱ZYW
Ý
(x 2 - x 1)2 + (y 2 - y 1)2
√
AB =
YW Ɂ YW
'IVEN
Step 1 Plot the points on a coordinate plane.
Step 2 Use the Distance Formula to find the
lengths of the sides of each triangle.
D=
Þ
XY Ɂ ZY
5/8/06
Given: D(-5, -5), E(-3, -1),
F(-2, -3), G(-2, 1),
H(0, 5), and I(1, 3)
Prove: ∠DEF ∠GHI
, EF = HI = √5
,
DE = GH = 2 √5
DF = GI = √
13 . Therefore DEF
GHI by SSS, and ∠DEF 1:03:57 PM
∠GHI by CPCTC.
Also available on transparency
INTERVENTION
Questioning Strategies
E X A M P LES 1 – 4
• What transformations are used to
change each triangle into the second congruent triangle?
2–3. See p. A16.
Lesson 4-6
261
Answers to Think and Discuss
−− −−
1. SAS; UW XZ;
∠U ∠X; ∠W ∠Z
THINK AND DISCUSS
−− −− −−− −−
1. In the figure, UV XY, VW YZ,
and ∠V ∠Y. Explain why
UVW XYZ. By CPCTC, which
additional parts are congruent?
2. See p. A4.
Critical Thinking
Emphasize that proofs
in this lesson go beyond
proving triangles congruent.
Encourage students to work backward when using CPCTC in proofs.
Remind students to first locate
which triangles they need to prove
congruent and then find the angles
or sides needed.
4-6 Exercises
1
8
6
7
9
<
2. GET ORGANIZED Copy and complete the graphic organizer.
Write all conclusions you can make using CPCTC.
̱
Ɂ ̱
*
/
4-6
Exercises
KEYWORD: MG7 4-6
KEYWORD: MG7 Parent
Assignment Guide
GUIDED PRACTICE
If you finished Examples 1–2
Basic 7–9, 17–19
Average 7–9, 17–20
Advanced 7–9, 17–20, 29, 32
SEE EXAMPLE
1
p. 260
If you finished Examples 1–4
Basic 7–19, 24–28, 33–37
Average 7–16, 20–28, 30,
33–37
Advanced 7–16, 19–37
Homework Quick Check
1. Vocabulary You use CPCTC after
proving triangles are congruent.
Which parts of congruent triangles
are referred to as corresponding parts?
Assign Guided Practice exercises
as necessary.
corr. and corr. sides
2. Archaeology An archaeologist
wants to find the height AB of a
rock formation. She places a
marker at C and steps off the
distance from C to B. Then she
walks the same distance from
C and places a marker at D.
If DE = 6.3 m, what is AB? 6.3 m
SEE EXAMPLE
p. 260
Quickly check key concepts.
Exercises: 7, 8, 10, 12, 14
2
−− −− −−
3. Given: X is the midpoint of ST. RX ⊥ ST
−− −−
Prove: RS RT
,
Proof:
262
-
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Chapter 4 Triangle Congruence
Answers
ge07se_c04_0260_0265.indd 262
and INTERVENTION
Diagnose Before the Lesson
4-6 Warm Up, TE p. 260
Monitor During the Lesson
Check It Out! Exercises, SE pp. 260–261
Questioning Strategies, TE p. 261
KEYWORD: MG7 Resources
262
Chapter 4
Assess After the Lesson
4-6 Lesson Quiz, TE p. 265
Alternative Assessment, TE p. 265
3a. Def. of ⊥
b. Rt. ∠ Thm.
c. Reflex. Prop. of d. Def. of mdpt.
e. RXS RXT
f. CPCTC
12/2/05 7:05:42 PM
SEE EXAMPLE
3
p. 261
−− −− −− −−
4. Given: AC AD, CB DB
−−
Prove: AB bisects ∠CAD.
?
−−−−
3. ACB ADB
2. b.
4. ∠CAB ∠DAB
−−
5. AB bisects ∠CAD
SEE EXAMPLE 4
p. 261
Reasons
a. Given
−− −−
b. AB AB
c. SSS Steps 1, 2
d. CPCTC
e. Def. of ∠ bisect
1. a. ?
−−−−
2. Reflex. Prop. of ?
−−−−
?
−−−−
5. e. ?
−−−−
3. c.
4. d.
Multi-Step Use the given set of points to prove each congruence statement.
5. E(-3, 3), F(-1, 3), G(-2, 0), J(0, -1), K(2, -1), L(1, 2); ∠EFG ∠JKL
6. A(2, 3), B(4, 1), C(1, -1), R(-1, 0), S(-3, -2), T(0, -4); ∠ACB ∠RTS
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
7. Surveying To find the distance AB across
a river, a surveyor first locates point C.
He measures the distance from C to B.
Then he locates point D the same distance
east of C. If DE = 420 ft, what is AB? 420 ft
A
Ê,,",
,/
Proof:
Statements
−− −−− −− −−
1. AC AD, CB DB
""
In Exercise 11, students may think
that if a ray bisects an angle of a
triangle then it also bisects the side
opposite the angle. Explain that this
is only true for some triangles.
Technology For Exercises
5, 6, 12, and 13, use
geometry software to
graph the points and to demonstrate
that the corresponding angle measures in each proof statement are
the same. Also address transformations after triangles are graphed.
Answers
5. EF = JK = 2, and EG = FG = JL
= KL = √
10 . So EFG JKL
7
1
500 ft
B
D
by SSS. ∠EFG ∠ JKL by CPCTC.
500 ft C
8–9
2
, BC = ST = √
10–11
3
13 ,
6. AB = RS = 2 √2
E
12–13
4
and RT = AC = √
17 . So ABC
−−− −− −− −−−
RST by SSS. ∠ ACB ∠RTS
8. Given: M is the midpoint of
9. Given: WX XY YZ ZW
Extra Practice
−−
−−
by
CPCTC.
PQ
and
RS.
Prove:
∠W
∠Y
Skills Practice p. S11
−− −−
−−
−−
Application Practice p. S31
Prove: QR PS
8. 1. M is the mdpt. of PQ and RS.
(Given)
7
8
,
−− −−− −− −−
2. PM QM, RM SM (Def. of
*
+
mdpt.)
3. ∠PMS ∠QMR (Vert. Thm.)
<
9
4. PMS QMR (SAS, Steps
−−
−−−
−− −−
2, 3)
10. Given: G is the midpoint of FH.
11. Given: LM bisects ∠JLK. JL KL
−− −−
−− −−
−−
5. QR PS (CPCTC)
EF EH
Prove: M is the midpoint of JK.
−− −− −− −−
9. 1. WX XY YZ ZW (Given)
Prove: ∠1 ∠2
−− −−
2. ZX ZX (Reflex. Prop. of )
3. WXZ YZX (SSS)
4. ∠W ∠Y (CPCTC)
−−
£
Ó
10.
1.
G is the mdpt. of FH. (Given)
2. FG = HG (Def. of mdpt.)
−− −−
3. FG HG (Def. of )
−−
Multi-Step Use the given set of points to prove each congruence statement.
4. Draw EG. (Through any 2 pts.
12. ST = VW = RT 12. R(0, 0), S(2, 4), T(-1, 3), U(-1, 0), V(-3, -4), W(-4, -1); ∠RST ∠UVW
there is exactly 1 line.)
−− −−
= UW = √
10 .
5.
EG
EG (Reflex. Prop. of )
13. A(-1, 1), B(2, 3), C(2, -2), D(2, -3), E(-1, -5), F(-1, 0); ∠BAC ∠EDF
−− −−
RS = UV = 2 √
5.
6. EF EH (Given)
−−
is adjacent to QTS. QS bisects ∠RQT. ∠R ∠T
So RST UVW 14. Given: QRS
7. EGF EGH (SSS Steps 3,
−−
−−
Prove: QS bisects RT.
by SSS. ∠RST 5, 6)
−−
−−
∠UVW by CPCTC. 15. Given: ABE and CDE with E the midpoint of AC
and BD
8.
∠EFG ∠EHG (CPCTC)
−− −−
Prove: AB CD
9. ∠1 ∠2 ( Supp. Thm.)
−−
11. 1. LM bisects ∠ JLK. (Given)
4-6 Triangle Congruence: CPCTC
263
2. ∠ JLM ∠KLM (Def. of ∠
bisect)
−− −−
−−
−−
3.
JL KL (Given)
AC
and
BD
.
(Given)
15.
1.
E
is
the
mdpt.
of
Answers
−− −−
−− −− −− −−
4. LM LM (Reflex. Prop. of )
2. AE CE; BE DE (Def. of mdpt.)
13. AB = DE = √
13 , BC = EF = 5, and
5. JLM KLM (SAS Steps 3,
3. ∠ AEB ∠CED (Vert. Thm.)
18 = 3 √
2 . So ABC ge07se_c04_0260_0265.indd 263AC = DF = √
12/2/05 7:05:46 PM
2, 4)
4. AEB CED (SAS Steps 2, 3)
−− −−
DEF by SSS. ∠BAC ∠EDF by CPCTC.
6. JM KM (CPCTC)
5.
∠
A
∠C
(CPCTC)
−−
−−
−−
−−
14. 1. QRS is adj. to QTS. QS bisects
7. M is the mdpt. of JK. (Def. of
6. AB CD (Conv. of Alt. Int. Thm.)
∠RQT. ∠R ∠T (Given)
mdpt.)
2. ∠RQS ∠TQS (Def. of ∠ bisect)
−− −−
3. QS QS (Reflex. Prop. of )
4. RSQ TSQ (AAS Steps 1, 2, 3)
−− −−
5. RS TS (CPCTC)
−−
−−
6. QS bisects RT. (Def. of bisect)
Lesson 4-6
263
Exercise 16 involves
using CPCTC to
prove corresponding sides of a triangle are congruent.
This exercise prepares students for
the Multi-Step Test Prep on page 280.
16. This problem will prepare you for the Multi-Step Test
Prep on page 280.
£äʈ˜° Óäʈ˜°
The front of a doghouse has the dimensions shown.
a. How can you prove that ADB ADC? HL
−− −−
b. Prove that BD CD.
−−
−−
c. What is the length of BD and BC to the nearest tenth?
Ê
Óäʈ˜°
17.3 in.; 34.6 in.
If students chose J
in Exercise 25, they
might be assuming
that the quadrilateral is a rectangle
with all angles congruent. If they
chose F or H, they might be assuming that the diagonals bisect the
angles.
Multi-Step Find the value of x.
17.
18.
14
ÝÊÊ££
ÓÝÊÊÎ
­ÈÝÊÊ{£®Â
Use the diagram for Exercises 19–21.
19. Given: PS = RQ, m∠1 = m∠4
Answers
−−
−−
16b.1. AD ⊥ BC (Given)
2. ∠ ADB and ∠ ADC are rt. .
(Def. of ⊥)
3. ADB and ADC are rt. .
(Def. of rt. )
4. AB = AC = 20 in. (Given)
−− −−
5. AB AC (Def. of )
−− −−
6. AD AD (Reflex. Prop. of )
7. ADB ADC (HL Steps
5, 6)
−− −−
8. BD = CD (CPCTC)
21
­{ÝÊÊ£®Â
*
Prove: m∠3 = m∠2
20. Given: m∠1 = m∠2, m∠3 = m∠4
£
-
Î
Ó
+
{
Prove: PS = RS
22. Yes; JKM
LMK by
21. Given: PS = RQ, PQ = RS
SSS, so ∠JKM −− −−
Prove: PQ RS
∠LMK by CPCTC.
−− −−
Therefore JK ML 22. Critical Thinking Does the diagram contain
enough information to allow you to conclude
by the Conv. of the
−− −−−
that JK ML? Explain.
Alt. Int. Thm.
,
23. Write About It Draw a diagram and explain how a surveyor can set up triangles
to find the distance across a lake. Label each part of your diagram. List which
sides or angles must be congruent.
19. 1. PS = RQ (Given)
−− −−
2. PS RQ (Def. of )
3. m∠1 = m∠4 (Given)
4. ∠1 ∠4 (Def. of )
−− −−
5. SQ QS (Reflex. Prop. of )
6. PSQ RQS (SAS Steps 2,
4, 5)
7. ∠3 ∠2 (CPCTC)
8. m∠3 = m∠2 (Def. of )
24. Which of these will NOT be used as a reason in a proof
−− −−
of AC AD?
SAS
ASA
CPCTC
Reflexive Property
25. Given the points K(1, 2), L(0, -4), M(-2, -3), and N(-1, 3),
which of these is true?
∠KNL ∠MNL
∠MLN ∠KLN
∠LNK ∠NLM
∠MNK ∠NKL
20. 1. m∠1 = m∠2, m∠3 = m∠4
(Given)
2. ∠1 ∠2, ∠3 ∠4 (Def. of )
−− −−
3. SQ SQ (Reflex. Prop. of )
4. PSQ RSQ (ASA Steps
2,3)
−− −−
5. PS RS (CPCTC)
26. What is the value of y?
10
20
6. PS = RS (Def. of )
35
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27. Which of these are NOT used to prove angles congruent?
21, 23. See p. A16.
4-6 PRACTICE A
congruent triangles
parallel lines
noncorresponding parts
perpendicular lines
4-6 PRACTICE C
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Chapter 4 Triangle Congruence
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28. Which set of coordinates represents the vertices of a
triangle congruent to RST? (Hint: Find the lengths
of the sides of RST.)
(3, 4), (3, 0), (0, 0)
(3, 3), (0, 4), (0, 0)
È
Þ
-
/
{
(3, 1), (3, 3), (4, 6)
(3, 0), (4, 4), (0, 6)
,
Ó
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ä
CHALLENGE AND EXTEND
Ó
{
È
,
32. ABC is in plane M. CDE is
in plane P. Both planes have
C in common and ∠A ∠E.
What is the height AB to the
nearest foot? 18 ft
−−
30. 1. Draw MK. (Through any 2 pts.
there is exactly 1 line.)
−− −−
2. KM MK (Reflex. Prop. of )
−− −− −− −−
3. JK LM, JM = LK (Given)
4. JKM = LMK (SSS Steps 2,
3)
5. ∠ J ∠L (CPCTC)
29. All of the edges of a cube are congruent. All of
29. Any diag. on
the angles on each face of a cube are right angles.
any face of the
Use
CPCTC
to
explain
why
any
two
diagonals
on
cube is the hyp.
−−
−−
the faces of a cube (for example, AC and AF )
of a rt. whose
must be congruent.
legs are edges of
the cube. Any 2
−−
−− −−− −− −−
of these are 30. Given: JK
ML, JM KL
31. Given: R is the midpoint of AB.
−−
by SAS. Therefore
Prove: ∠J ∠L
S is the midpoint of DC.
−− −−
RS ⊥ AB, ∠ASD ∠BSC
any 2 diags. are (Hint: Draw an auxiliary line.)
by CPCTC.
Prove: ASD BSC
Answers
-
Journal
Have students explain what CPCTC
means. They should include an
example with their explanation.
Have students work in small groups
to create an example where they
must first prove triangles congruent
in order to reach a required
conclusion.
4-6
£äÊvÌ
Ó£ÊvÌ 1. Given: Isosceles PQR,
−− −− −−
base QR, PA PB
−− −−
Prove: AR BQ
*
SPIRAL REVIEW
33. Lina’s test scores in her history class are 90, 84, 93, 88, and 91. What is the minimum
score Lina must make on her next test to have an average test score of 90?
(Previous course) 94
35. reflection
across the x-axis
34. One long-distance phone plan costs $3.95 per month plus $0.08 per minute of use.
A second long-distance plan costs $0.10 per minute for the first 50 minutes used
each month and then $0.15 per minute after that. Which plan is cheaper if you use
an average of 75 long-distance minutes per month? (Previous course)
36. translation
(x, y) →
(x - 3, y - 4)
The second plan is cheaper.
A figure has vertices at (1, 3), (2, 2), (3, 2), and (4, 3). Identify the transformation of
the figure that produces an image with each set of vertices. (Lesson 1-7)
37. Yes; it is given
that ∠B
∠D
−− −−
and BC DC. By
the Vert. ∠ Thm.,
∠BCA ∠DCE.
Therefore ABC EDC by ASA.
35.
(1, -3), (2, -2), (3, -2), (4, -3)
36.
(-2, -1), (-1, -2), (0, -2), (1, -1)
37. Determine if you can use ASA to prove
ACB ECD. Explain. (Lesson 4-5)
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−−
1. Isosc. PQR, base QR (Given)
−− −−
2. PQ = PR (Def. of Isosc. )
−− −−
3. PA = PB (Given)
4. ∠P ∠P (Reflex. Prop. of )
5. QPB RPA (SAS Steps
2, 4, 3)
−− −−
6. AR = BQ (CPCTC)
+
2. Given: X is the midpoint of
−−
AC. ∠1 ∠2
−−
Prove: X is the midpoint of BD.
£
8
Ó
−−
1. X is mdpt. of AC. ∠1 ∠2
(Given)
4-6 Triangle Congruence: CPCTC
265
2. AX = CX (Def. of mdpt.)
−− −−
3. AX CX (Def. of )
Answers
4. ∠ AXD ∠CXB (Vert. Thm.)
−−
5. AXD CXB (ASA Steps
31. 1. R is the mdpt. of AB. (Given)
−− −−
1, 4, 5)
2. AR BR (Def. of mdpt.)
12/2/05 7:05:55 PM
−− −−
−−
−−
6. DX BX (CPCTC)
3. RS ⊥ AB (Given)
7. DX = BX (Def. of )
4. ∠ ARS and ∠BRS are rt. . (Def. of ⊥)
−−
8. X is mdpt. of BD. (Def. of mdpt.)
5. ∠ ARS ∠BRS (Rt. ∠ Thm.)
−− −−
6. RS RS (Reflex. Prop. of )
3. Use the given set of points to
7. ARS BRS (SAS Steps 2, 5, 6)
−− −−
prove DEF GHJ: D(–4,
8. AS BS (CPCTC)
4),
E(–2, 1), F(–6, 1), G(3, 1),
9. ∠ ASD ∠BSC (Given)
−−
H(5,
–2), J(1, –2).
10. S is the mdpt. of DC. (Given)
−− −−
DE
=
GH = √
13 , DF = GJ
11. DS = CS (Def. of mdpt.)
= √
13 , EF = HJ = 4, and
12. ASD BSC (SAS Steps 8, 9, 11)
DEF GHJ by SSS.
4WOTRIANGULARPLATESARECONGRUENT4HEAREAOFONEOFTHEPLATESISSQUAREINCHES
7HATISTHEAREAOFTHEOTHERPLATE%XPLAIN
Also available on transparency
Lesson 4-6
265
Organizer
Quadratic Equations
See Skills Bank
page S66
Algebra
1
Traditional __
day
2
1
__
Block day
4
See Skills Bank
page S66
Objective: Solve quadratic
GI
equations to find the length of a
side of a triangle.
@<I
<D
A quadratic equation is an equation that can be written in the
form ax 2 + bx + c = 0.
Algebra
Pacing:
Example
−− −−
Given: ABC is isosceles with AB AC. Solve for x.
Online Edition
Ý ÓÊÊxÝ
Step 1 Set x 2 – 5x equal to 6 to get x 2 – 5x = 6.
Countdown to
Testing Week 9
Step 2 Rewrite the quadratic equation by subtracting 6
from each side to get x 2 – 5x – 6 = 0.
Step 3 Solve for x.
Method 1: Factoring
Teach
x - 5x - 6 = 0
(x - 6)(x + 1) = 0
Students review and apply the methods of solving quadratic equations.
x=6
For addiINTERVENTION
tional review and practice on factoring and using the Quadratic Formula,
see Skills Bank page S66.
Critical Thinking Have
students check each solution. Even if the value of x
is negative, a length of the side
cannot be negative.
or
Set each factor
equal to 0.
x = -1
Solve.
Step 4 Check each solution in the original equation.
x 2 - 5x = 6
x 2 - 5x = 6
2
(6 ) - 5 (6 )
6
(-1) - 5 (-1 )
6
36 - 30
6
1+5
6
6
6
Close
2
Assess
Have students compare and contrast
the two methods used for finding
the value of the variable.
6
Solve for x in each isosceles triangle.
−− −−
1. Given: FE FG
−− −−
2. Given: JK JL
Ý ÓÊÊÎÝ
£n
−− −−
3. Given: YX YZ
6 or -2
266
Ý ÓÊÊ{Ý
9
−− −−
4. Given: QP QR
£Ó
<
Ý ÓÊÊ{Ý
-3 or 1
,
£Ó
Î
+
Ý ÓÊÊÓÝ
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Chapter 4 Triangle Congruence
ge07se_c04_0266.indd 266
Chapter 4
-6 or 2
8
266
6 Try This
6 or -3
KEYWORD: MG7 Resources
-b ± √
b 2 - 4ac
x = __
2a
-(-5) ± √(
-5)2 - 4(1)(-6)
Substitute 1 for
x = ___
a, -5 for b,
2(1)
and -6 for c.
√
5
±
49
x=_
Simplify.
2
5±7
x=_
Find the square root.
2
12 or x = _
-2
x=_
Simplify.
2
2
x = 6 or x = -1
Factor.
x - 6 = 0 or x + 1 = 0
Method 2: Quadratic Formula
2
Remember
È
9/6/05 9:39:33 AM
4-7
Introduction to
Coordinate Proof
4-7
Organizer
Pacing: Traditional 2 days
Block 1 day
Prove geometric
concepts by using
coordinate proof.
the coordinate plane for use in
coordinate proofs.
Prove geometric concepts by using
coordinate proof.
You have used coordinate geometry
to find the midpoint of a line segment
and to find the distance between
two points. Coordinate geometry can
also be used to prove conjectures.
GI
Vocabulary
coordinate proof
Objectives: Position figures in
Who uses this?
The Bushmen in South Africa use
the Global Positioning System to
transmit data about endangered
animals to conservationists.
(See Exercise 24.)
Objectives
Position figures in the
coordinate plane for use
in coordinate proofs.
<D
@<I
Online Edition
Tutorial Videos
Countdown to
Testing Week 9
A coordinate proof is a style of proof that uses coordinate geometry
and algebra. The first step of a coordinate proof is to position the given
figure in the plane. You can use any position, but some strategies can
make the steps of the proof simpler.
Warm Up
Strategies for Positioning Figures in the Coordinate Plane
• Use the origin as a vertex, keeping the figure in Quadrant I.
Evaluate.
• Center the figure at the origin.
1. Find the midpoint between
(0, 2x) and (2y, 2z). (y, x + z)
• Center a side of the figure at the origin.
• Use one or both axes as sides of the figure.
EXAMPLE
1
2. One leg of a right triangle has
length 12, and the hypotenuse
has length 13. What is the
length of the other leg? 5
Positioning a Figure in the Coordinate Plane
Position a rectangle with a length of 8 units and a width of 3 units in the
coordinate plane.
Method 1 You can center the longer
side of the rectangle at the origin.
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Method 2 You can use the origin as
a vertex of the rectangle.
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3. Find the distance between
(0, a) and (0, b), where b > a.
b-a
Ó
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Ý
ä ­ä]Êä®
{
È
­n]Êä®
Depending on what you are using the figure to prove, one solution may
be better than the other. For example, if you need to find the midpoint
of the longer side, use the first solution.
A: A rectangle!
Answers to Check it Out!
1. Position a right triangle with leg lengths of 2 and 4 units in
the coordinate plane. (Hint: Use the origin as the vertex of
the right angle.)
4- 7 Introduction to Coordinate Proof
Q: What do you call a broken
angle?
1. See p. A16.
267
1 Introduce
ge07se_c04_0267_0272.indd 267
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Motivate
12/3/05 7:04:49 PM
Relate coordinate proofs to other methods of
proofs students have used. Point out that they will
use algebra and what they already know about
triangles in the coordinate plane in this geometric proof. Make transparencies of graph paper so
students can display their work on the overhead
projector as they discuss coordinate proofs.
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Explorations and answers are provided in the
Explorations binder.
KEYWORD: MG7 Resources
Lesson 4-7
267
Once the figure is placed in the coordinate plane, you can use slope, the
coordinates of the vertices, the Distance Formula, or the Midpoint Formula
to prove statements about the figure.
Additional Examples
Example 1
EXAMPLE
2
Position a square with a side
length of 6 units in the coordinate plane. Possible answer:
­ä]ÊÈ®
y
Write a coordinate proof.
Given: Right ABC has vertices A(0, 6),
B(0, 0), and C(4, 0). D is the
−−
midpoint of AC.
Prove: The area of DBC is one half the
area of ABC.
­È]ÊÈ®
­È]Êä®
{
Ó
Ý
Ó
È
(
of DBC, and the base is 4 units.
area of DBC = __12 bh
= __12 (4)(3) = 6 square units
Since 6 = _12_(12), the area of DBC is one half the area of ABC.
2. Use the information in Example 2 to write a coordinate proof
showing that the area of ADB is one half the area of ABC.
A coordinate proof can also be used to prove that a certain relationship is
always true. You can prove that a statement is true for all right triangles
without knowing the side lengths. To do this, assign variables as the coordinates
of the vertices.
Example 3
Position each figure in the
coordinate plane and give the
coordinates of each vertex.
EXAMPLE
A. rectangle with width m and
length twice the width
Possible answer:
3
­Ó“]ʓ®
x
­ä]Êä®
­Ó“]Êä®
Assigning Coordinates to Vertices
Position each figure in the coordinate plane and give the coordinates of
each vertex.
y
­ä]ʓ®
A a right triangle with leg
Do not use both axes
when positioning
a figure unless you
know the figure
has a right angle.
B. right triangle with legs of
lengths s and t
Possible answer:
lengths a and b
Þ
Also available on transparency
268
• How do you decide which strategy to use to position a figure in
a coordinate plane? What are the
advantages of each position?
Answers to Check it Out!
2–3. See p. A16.
Chapter 4
Þ
­V]Ê`®
Ý
­L]Êä®
Ý
­ä]Êä®
­V]Êä®
If a coordinate proof requires calculations with fractions, choose coordinates
that make the calculations simpler. For example, use multiples of 2 when
you are to find coordinates of a midpoint. Once you have assigned the
coordinates of the vertices, the procedure for the proof is the same, except
that your calculations will involve variables.
­Ì]Êä®
E X AM P LES 1 – 3
­ä]Ê`®
3. Position a square with side length 4p in the coordinate plane
and give the coordinates of each vertex.
x
Questioning Strategies
B a rectangle with
length c and width d
­ä]Ê>®
­ä]Êä®
y
INTERVENTION
)
0 + 4 ____
D = ____
, 6 +2 0 = (2, 3). The y-coordinate of D is the height
2
Given: Rectangle ABCD with
A(0, 0), B(4, 0), C(4, 10),
and D(0, 10)
Prove: The diagonals bisect each
other.
−−
−−
Mdpt. of AC is (2, 5). Mdpt. of BD
is also (2, 5). Therefore the diags.
bisect each other.
268
By the Midpoint Formula, the coordinates of
Write a coordinate proof.
­ä]Êä®
È
= __12 (4)(6) = 12 square units
Example 2
­ä]Êî
Þ
Proof: ABC is a right triangle with height
AB and base BC.
area of ABC = __12 bh
x
­ä]Êä®
Writing a Proof Using Coordinate Geometry
Chapter 4 Triangle Congruence
2 Teach
ge07se_c04_0267_0272.indd 268
12/3/05 7:04:56 PM
Guided Instruction
Discuss how to position figures in the coordinate plane. Show students how to assign
convenient variable coordinates to vertices.
Have students review the Distance and
Midpoint Formulas before using them in
coordinate proofs.
Social Studies René Descartes,
a 17th century mathematician
and philosopher, first developed
the coordinate plane to make it easier to
describe the position of objects.
Through Multiple Representations
Divide the class into two groups. Have one
group complete a proof that a triangle with
coordinates (0, 4), (0, 0), and (3, 0) is a
right triangle. Have the second group use
the coordinates (0, a), (0, 0), and (b, 0)
and then verify that the triangle is a right
triangle. Compare and contrast the two
methods of proof. Emphasize that when
you use variables, you prove that the statement is true for all right triangles, not just
for a specific triangle.
EXAMPLE
4
Writing a Coordinate Proof
""
−−
Given: ∠B is a right angle in ABC. D is the midpoint of AC.
Prove: The area of DBC is one half the area of ABC.
Step 1 Assign coordinates to each vertex.
The coordinates of A are (0, 2 j),
the coordinates of B are (0, 0),
and the coordinates of C are (2n, 0).
­ä]ÊӍ®
Þ
Step 3 Write a coordinate proof.
Because the x- and
y-axes intersect at
right angles, they
can be used to form
the sides of a right
triangle.
In Example 2, students may want to
−−
−−
use AC as the base and BD as the
height. Remind students that in a
right triangle, either leg can be the
base and the other leg the height.
Since you will use the
Midpoint Formula to find the
coordinates of D, use multiples
of 2 for the leg lengths.
Step 2 Position the figure in the coordinate plane.
Ý
Proof: ABC is a right triangle with height 2j
and base 2n.
Ê,,",
,/
­ä]Êä®
­Ó˜]Êä®
Algebra Remind students
that when finding the midpoint, they can simplify by
factoring. For example,
2(a + b)
2a + 2b
______
= _______ = a + b.
area of ABC = __12 bh
2
= __12 (2n)(2j)
Inclusion Point out
that students are not to
choose coordinates that
are “special” when doing proofs. For
example, they should not choose the
vertices of a rectangle so that it is a
square. They should use coordinates
that make the figure a nonsquare
rectangle.
= 2nj square units
(
2
)
2j + 0
0 + 2n _____
By the Midpoint Formula, the coordinates of D = _____
, 2
= (n, j).
2
The height of DBC is j units, and the base is 2n units.
area of DBC = _12_bh
= _12_(2n)(j)
= nj square units
Since nj = _1_(2nj), the area of DBC is one half the area of ABC.
2
4. Use the information in Example 4 to write a coordinate
proof showing that the area of ADB is one half the
area of ABC.
Additional Examples
Example 4
Given: Rectangle PQRS
Prove: The diagonals are .
y
THINK AND DISCUSS
+
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­ä]ÊL® *
1. When writing a coordinate proof why are variables used instead of
numbers as coordinates for the vertices of a figure?
x
2. How does the way you position a figure in the coordinate plane affect
your calculations in a coordinate proof?
2
PR = √a
+ b 2 , and QS
2
= √
a + b 2 . Thus the diagonals
are .
3. Explain why it might be useful to assign 2p as a coordinate instead
of just p.
4. GET ORGANIZED Copy and complete the graphic organizer.
In each row, draw an example of each strategy that might be used
when positioning a figure for a coordinate proof.
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4
• How is the figure placed in the
coordinate plane? Why?
4- 7 Introduction to Coordinate Proof
Answers to Check it Out!
3 Close
ge07se_c04_0267_0272.indd 269
Summarize
Review the strategies for positioning figures
in the coordinate plane. Remind students
to choose coordinates that make calculations simpler. Advise students to use variables instead of numbers as coordinates
for the vertices of a figure when doing a
proof.
269
4. See p. A16.
and INTERVENTION
12/3/05 7:05:01 PM
Answers to Think and Discuss
Diagnose Before the Lesson
4-7 Warm Up, TE p. 267
1. Possible answer: By using variables,
your results are not limited to specific
numbers.
Monitor During the Lesson
Check It Out! Exercises, SE pp. 267–269
Questioning Strategies, TE pp. 268–269
2. Possible answer: The way you position the figure will affect the coords.
assigned to the vertices and therefore
your calculations.
Assess After the Lesson
4-7 Lesson Quiz, TE p. 272
Alternative Assessment, TE p. 272
3. Possible answer: If you need to calculate the coords. of a mdpt., assigning
2p allows you to avoid using fractions.
4. See p. A4.
Lesson 4-7
269
4-7 Exercises
4-7
Exercises
KEYWORD: MG7 4-7
KEYWORD: MG7 Parent
GUIDED PRACTICE
Assignment Guide
Assign Guided Practice exercises
as necessary.
If you finished Examples
Basic 8–10, 15–17,
Average 8–10, 15–17,
Advanced 8–10, 15–17,
1–2
21
20–22
20–24
If you finished Examples 1–4
Basic 8–13, 15–21, 26–30,
35–40
Average 8–15, 21, 22, 25–32,
35–40
Advanced 8–15, 21–40
1. Vocabulary What is the relationship between coordinate geometry,
coordinate plane, and coordinate proof ? Possible answer: In coord. geometry, a coord.
SEE EXAMPLE
1
p. 267
3. a right triangle with leg lengths of 1 unit and 3 units
SEE EXAMPLE
2
SEE EXAMPLE
3
p. 268
x
3.
y
x
Ó
Ý
Position each figure in the coordinate plane and give
the coordinates of each vertex.
,
SEE EXAMPLE 4
p. 269
Independent Practice
For
See
Exercises Example
8–9
10
11–12
13
1
2
3
4
+
È
Multi-Step Assign coordinates to each vertex and write a coordinate proof.
−−
7. Given: ∠R is a right angle in PQR. A is the midpoint of PR.
−−
B is the midpoint of QR.
Prove: AB = __12 PQ
Position each figure in the coordinate plane.
8. a square with side lengths of 2 units
9. a right triangle with leg lengths of 1 unit and 5 units
Extra Practice
Skills Practice p. S11
5. Possible answer:
Application Practice p. S31
Write a proof using coordinate geometry.
Þ
10. Given: Rectangle ABCD has coordinates A(0, 0),
B(0, 10), C(6, 10), and D(6, 0). E is the
−−
−−
midpoint of AB, and F is the midpoint of CD.
Prove: EF = BC
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m
Position each figure in the coordinate plane and give
the coordinates of each vertex.
x
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n
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y
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6. a rectangle with length a and width b
2. Possible answer:
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*
5. a right triangle with leg lengths m and n
Answers
Write a proof using coordinate geometry.
4. Given: Right PQR has coordinates P(0, 6), Q(8, 0),
−−
and R(0, 0). A is the midpoint of PR.
−−
B is the midpoint of QR.
Prove: AB = __12 PQ
p. 268
Homework Quick Check
Quickly check key concepts.
Exercises: 8, 10, 12, 13, 15
proof is a proof in which you place figures in
Position each figure in the coordinate plane. the coord. plane to prove a result.
2. a rectangle with a length of 4 units and width of 1 unit
4, 6–13. See pp. A16–A17.
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ä
12. a rectangle with dimensions x and 3x
Ó
{
È
Multi-Step Assign coordinates to each vertex and write a coordinate proof.
−−
−−
13. Given: E is the midpoint of AB in rectangle ABCD. F is the midpoint of CD.
Prove: EF = AD
14. Critical Thinking Use variables to write the general form of the endpoints
of a segment whose midpoint is (0, 0). (x, y) and (-x, -y)
270
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15. Recreation A hiking trail begins at E(0, 0). Bryan hikes from the start of the trail to
a waterfall at W (3, 3) and then makes a 90° turn to a campsite at C(6, 0).
""
a. Draw Bryan’s route in the coordinate plane.
In Exercise 19, students may give
the missing coordinates as (0, p)
instead of (p, 0). Review with students the order of coordinates.
b. If one grid unit represents 1 mile, what is the total distance Bryan hiked?
Round to the nearest tenth. 8.5 mi
Find the perimeter and area of each figure.
(
√
5 ) units; a 2 square units
16. a right triangle with leg lengths of a and 2a units a 3 +
17. a rectangle with dimensions s and t units 2s + 2t units; st square units
Find the missing coordinates for each figure.
18.
(n, n)
Þ
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Conservation
19.
Þ
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Exercise 26
involves using
the Pythagorean
Theorem to find the length of a
missing side of a right triangle. This
exercise prepares students for the
Multi-Step Test Prep on page 280.
(p, 0)
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Ý
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,/
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Answers
20. Conservation The Bushmen have sighted animals at the following coordinates:
(-25, 31.5), (-23.2, 31.4), and (-24, 31.1). Prove that the distance between two of
these locations is approximately twice the distance between two other.
The origin of the
springbok’s name may
come from its habit of
pronking, or bouncing.
When pronking, a
springbok can leap up
to 13 feet in the air.
Springboks can run up
to 53 miles per hour.
15a.
21. Navigation Two ships depart from a port at P(20, 10). The first ship travels to
a location at A(-30, 50), and the second ship travels to a location at B(70, -30).
Each unit represents one nautical mile. Find the distance to the nearest nautical
mile between the two ships. Verify that the port is at the midpoint between the two.
x
20.
22. Given: Rectangle PQRS has coordinates P(0, 2), Q(3, 2), R (3, 0), and S(0, 0).
−−
−−
PR and QS intersect at T (1.5, 1).
Prove: The area of RST is __14 of the area of the rectangle.
(
(-25 + 23.2)2 + (31.5 - 31.4) 2
√
≈ 1.8
+ 24)2 + (31.4 - 31.1) 2
√(-23.2
≈ 0.9
(-24 + 25) 2 + (31.1 - 31.5) 2
√
≈ 1.1
Write a coordinate proof.
y +y
x 1 + x 2 _____
, 12 2
23. Given: A(x 1, y 1), B(x 2, y 2), with midpoint M _____
2
Prove: AM = _12_AB
y
)
1.8 is twice 0.9. The dist.
between 2 of the locations is
approximately twice the dist.
between 2 of the other.
24. Plot the points on a coordinate plane and connect them to form KLM and
MPK. Write a coordinate proof.
Given: K (-2, 1), L(-2, 3), M(1, 3), P(1, 1)
Prove: KLM MPK
21. AB ≈ 128 nautical miles;
AP = BP ≈ 64 nautical miles; so
−−
P is the mdpt. of AB.
25. Write About It When you place two sides of a figure on the coordinate axes,
what are you assuming about the figure? You are assuming the figure has a rt. ∠.
26. This problem will prepare you for the Multi-Step Test Prep on page 280.
Þ
Paul designed a doghouse to fit against the side of his house.
His plan consisted of a right triangle on top of a rectangle.
a. Find BD and CE. BD = 38 in.; CE = 24 in.
b. Before building the doghouse, Paul sketched his plan
on a coordinate plane. He placed A at the origin
−−
and AB on the x-axis. Find the coordinates of B, C, D,
and E, assuming that each unit of the coordinate
plane represents one inch.
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1
1
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Lesson 4-7
271
For Exercise 28, discuss that the vertices
of a rectangle do not
have to be in the first quadrant.
27. The coordinates of the vertices of a right triangle are (0, 0), (4, 0), and (0, 2).
Which is a true statement?
The vertex of the right angle is at (4, 2).
The midpoints of the two legs are at (2, 0) and (0, 1).
units.
The hypotenuse of the triangle is √6
The shortest side of the triangle is positioned on the x-axis.
If students chose F for Exercise 30,
they found the midpoint between A
and C, not the midpoint between A
and B.
28. A rectangle has dimensions of 2g and 2f units. If one vertex is at the origin,
which coordinates could NOT represent another vertex?
(2f, 0)
(2f, g)
(2g, 2f)
(-2f, 2g)
29. The coordinates of the vertices of a rectangle are (0, 0), (a, 0), (a, b), and (0, b).
What is the perimeter of the rectangle?
1 ab
_
a+b
ab
2a + 2b
2
30. A coordinate grid is placed over a map. City A is located at (-1, 2) and city C is
located at (3, 5). If city C is at the midpoint between city A and city B, what are
the coordinates of city B?
(1, 3.5)
(7, 8)
(2, 7)
(-5, -1)
Journal
Ask students to explain the strategies they use for positioning a figure
in the coordinate plane. Then have
them support their explanation with
a drawing of a geometric shape in
the coordinate plane.
CHALLENGE AND EXTEND
Find the missing coordinates for each figure.
31.
Þ
(a + c, b)
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32.
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(p, 0)
Ý
Have students place a rectangle in a
coordinate plane so that the length
is twice the width. Then ask them to
make a conjecture about the area of
the rectangle and to write a coordinate proof.
33. Possible answer:33.
Rotate the 180°
and translate it
34.
vertically 2s units.
The new coords.
would be (0, 0),
(0, 2s), and (2s, 0).
4-7
The vertices of a right triangle are at (-2s, 2s), (0, 2s), and (0, 0). What coordinates
could be used so that a coordinate proof would be easier to complete?
Rectangle ABCD has dimensions of 2f and 2g units.
g
−−
The equation of the line containing BD is y = __f x, and
−−
g
the equation of the line containing AC is y = - __f x + 2g.
39. y 22
272
­x>]Êä®
34.
_ x = - _ x + 2g
g
g
f
2g
f
ge07se_c04_0267_0272.indd 272
3. Given: Rectangle ABCD with
coordinates A(0, 0), B(0, 8), C
(5, 8), and D(5, 0). E is mdpt.
−−
−−
of BC, and F is mdpt. of AD.
Prove: EF = AB
By the Midpoint Formula, the
5
coordinates of E are __
,8
2
5
__
and F are 2 , 0 .
Then EF = 8, and AB = 8.
Thus EF = AB.
( )
Also available on transparency
37. 0 = 3x 2 - x - 10
2 or -1.67
ÈnÂ
Chapter 4 Triangle Congruence
Answers
x
1.19 or -4.19
40. Use A(-4, 3), B(-1, 3), C (-3, 1), D(0, -2), E(3, -2), and F (2, -4) to prove
∠ABC ∠EDF. (Lesson 4-6).
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y
272
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38. x 112
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( )
Find each value. (Lesson 3-2)
­È]Êή
2. square with side lengths of
5a units. Possible answer:
36. 0 = x 2 + 3x - 5
-2.5 or 0.25
x
­ä]Êä®
­Óv]ÊÓ}®
Þ
Ý
Use algebra to show that the coordinates of E are (f, g).
35. 0 = 8x 2 + 18x - 5
y
­ä]Êx>®
­ä]ÊÓ}®
Use the quadratic formula to solve for x. Round to the nearest
hundredth if necessary. (Previous course)
1. rectangle with a length of
6 units and a width of 3 units
Possible answer:
­ä]Êä®
­ÊÊÊ]ÊÊÊÊ®
SPIRAL REVIEW
Position each figure in the
coordinate plane.
­ä]Êή
­ä]Êä®
_ x = 2g
f
x=f
g
y= x
f
g
y = (f)
f
y=g
_
_
Set eqns. = to
each other.
Combine like
terms.
Simplify.
Given
Subst.
Simplify.
, AC = √
40. AB = 3, BC = 2 √2
5
, EF = √
ED = 3, DF = 2 √2
5
−− −−
−− −− −− −−
AB ED, BC DF, and AC EF.
Therefore ABC EDF by SSS,
and ∠ ABC ∠EDF by CPCTC.
12/3/05 7:05:23 PM
Isosceles and
Equilateral Triangles
4-8
4-8
Organizer
Pacing: Traditional 1 day
Block
Apply properties of
isosceles and equilateral
triangles.
about isosceles and equilateral
triangles.
Apply properties of isosceles and
equilateral triangles.
GI
Recall that an isosceles triangle has at least
two congruent sides. The congruent sides
are called the legs . The vertex angle is
the angle formed by the legs. The side
opposite the vertex angle is called the base ,
and the base angles are the two angles
that have the base as a side.
Vocabulary
legs of an isosceles
triangle
vertex angle
base
base angles
<D
HYPOTHESIS
Converse of Isosceles
Triangle Theorem
True or False. If false explain.
If two angles of a triangle are
congruent, then the sides opposite
those angles are congruent.
1. Find each angle
measure.
60°; 60°; 60°
∠B ∠C
2. Every equilateral triangle is
isosceles. T
−− −−
DE DF
Theorem 4-8-1 is proven below. You will prove Theorem 4-8-2 in Exercise 35.
Isosceles Triangle Theorem
−− −−
Given: AB AC
Prove: ∠B ∠C
Proof:
1.
2.
3.
4.
5.
Statements
The Isosceles
Triangle Theorem is
sometimes stated as
“Base angles of an
isosceles triangle
are congruent.”
3. Every isosceles triangle is
equilateral. F; an isosc. has only 2 sides.
Also available on transparency
8
Warm Up
CONCLUSION
Isosceles Triangle Theorem
If two sides of a triangle are
congruent, then the angles opposite
the sides are congruent.
PROOF
Tutorial Videos, Interactivity
Isosceles Triangle
THEOREM
4-8-2
Online Edition
Î
£
4-8-1
@<I
Countdown to
Testing Week 9
Ó
∠3 is the vertex angle.
∠1 and ∠2 are the
base angles.
Theorems
2
Objectives: Prove theorems
Who uses this?
Astronomers use geometric methods.
(See Example 1.)
Objectives
Prove theorems about
isosceles and equilateral
triangles.
__1 day
−−
Draw X, the mdpt. of BC.
−−
Draw the auxiliary line AX.
−− −−
BX CX
−− −−
AB AC
−− −−
AX AX
Reasons
1. Every seg. has a unique mdpt.
2. Through two pts. there is exactly one line.
3. Def. of mdpt.
A: “I’ve got more problems than a
math book.”
4. Given
5. Reflex. Prop. of 6. ABX ACX
6. SSS Steps 3, 4, 5
7. ∠B ∠C
7. CPCTC
4-8 Isosceles and Equilateral Triangles
Q: What did the frowning student
say to his teacher?
273
1 Introduce
ge07se_c04_0273_0279.indd 273
E X P L O R AT I O N
Motivate
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12/9/05 12:44:23 PM
Give small groups of students different examples
of isosceles and equilateral triangles. Have each
group measure the lengths of the three sides and
the three angles of their triangles. Then ask them
to make as many conjectures about isosceles and
equilateral triangles as they can. For example, students should recognize that the base angles of an
isosceles triangle are congruent.
3TATEACONJECTUREBASEDONYOURRESULTS
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Explorations and answers are provided in the
Explorations binder.
*
KEYWORD: MG7 Resources
Lesson 4-8
273
EXAMPLE
1
Additional Examples
Example 1
−−
The length of YX is 20 feet.
−−
Explain why the length of YZ is
the same.
9
ÓäÊvÌ
<
{äÂ
−− −−
m∠YZX = 40°, so XY YZ by
Conv. of Isosc. Thm. Thus YZ
= 20 ft.
EXAMPLE
2
­ÝÊ{{®Â
ÎnÂ
Isosc. Thm.
ÝÂ
Sum Thm.
Simplify and subtract 38 from both sides.
Divide both sides by 2.
,
/
ÓÝÂ
*
ÈÞÂ
­nÞÊʣȮÂ
{nÂ
48°
Questioning Strategies
­ÝÊÊÎä®Â
Find each angle measure.
2a. m∠H
2b. m∠N
66°
Substitute the given values.
Isosc. Thm.
m∠S = m∠R
Substitute the given values.
2x° = (x + 30)°
Subtract x from both sides.
x = 30
Thus m∠S = 2x° = 2(30) = 60°.
INTERVENTION
The following corollary and its converse show the connection between
equilateral triangles and equiangular triangles.
1
• How is the Converse of the
Isosceles Triangle Theorem used?
E X AM P LE
*
Not drawn to scale
B m∠S
Also available on transparency
E X AM P LE
-՘
Finding the Measure of an Angle
m∠C = m∠B = x°
m∠C + m∠B + m∠A = 180
x + x + 38 = 180
2x = 142
x = 71
Thus m∠C = 71°.
ÓÓÂ
ÎÝÂ
™ä°{¨
>˜Õ>ÀÞ
A m∠C
B. m∠G
66°
ՏÞ
Find each angle measure.
Find each angle measure.
n™°È¨
1. If the distance from Earth to a star in September is 4.2 × 10 13 km,
what is the distance from Earth to the star in March? Explain.
Example 2
A. m∠F
79°
m∠LKM = 180 - 90.4, so m∠LKM = 89.6°. Since ∠LKM ∠M,
LMK is isosceles by the Converse of the Isosceles Triangle Theorem.
Thus LK = LM = 4.0 × 10 13 km.
£{äÂ
8
Astronomy Application
The distance from Earth to nearby
stars can be measured using the
parallax method, which requires
observing the positions of a star
6 months apart. If the distance LM
to a star in July is 4.0 × 10 13 km,
explain why the distance LK to the
star in January is the same. (Assume
the distance from Earth to the Sun
does not change.)
Corollary 4-8-3
Equilateral Triangle
COROLLARY
2
• When you find the value of x, is
this the answer to the problem?
Explain.
HYPOTHESIS
If a triangle is equilateral, then it is
equiangular.
(equilateral → equiangular )
CONCLUSION
∠A ∠B ∠C
You will prove Corollary 4-8-3 in Exercise 36.
274
Chapter 4 Triangle Congruence
Answers to Check It Out!
2 Teach
Guided Instruction
ge07se_c04_0273_0279.indd 274
Have students practice finding angle measures of isosceles and equilateral triangles
using natural numbers before using algebraic expressions. Review the most efficient
way to name the vertices of a triangle
before doing a coordinate proof.
Through Visual Cues
Represent the Isosceles Triangle Theorem as:
If
then
and the Converse of Isosceles Triangle
Theorem as:
If
274
Chapter 4
then
1. 4.2 × 10 13; since there are 6 months
between September and March, the
∠ measures will be approximately the
same between Earth and the star. By
the Conv. of the Isosc. Thm., the created are isosc., and the dist. is the
same.
12/20/05 3:21:56 PM
Corollary 4-8-4
Equiangular Triangle
COROLLARY
HYPOTHESIS
CONCLUSION
Additional Examples
If a triangle is equiangular, then it is
equilateral.
Example 3
−− −− −−
DE DF EF
(equiangular → equilateral )
Find each value.
A. x
14
You will prove Corollary 4-8-4 in Exercise 37.
­ÓÝÊÎÓ®Â
EXAMPLE
3
Using Properties of Equilateral Triangles
Find each value.
A x
ABC is equiangular.
(3x + 15)° = 60°
3x = 45
x = 15
B. y
Equilateral → equiangular "
Subtract 15 from both sides.
*
ÓÌÊÊ£
Def. of equilateral Subtract 2t and add 8 to
both sides.
{ÌÊÊn
Divide both sides by 2.
3. Use the diagram to find JL. 10
4. By the Mdpt.
Formula, the coords.
of X are (-a, b), the
coords. of Y are (a, b),
and the coords. of Z
are (0, 0). By the Dist.
Formula, XZ = YZ
−−
= √
a 2 + b 2 . So XZ
−−
YZ and XYZ
is isosc.
{ÞÊ£Ó
Equiangular → equilateral t = 4.5
A coordinate proof
may be easier if you
place one side of the
triangle along the
x-axis and locate a
vertex at the origin
or on the y-axis.
xÞÊÈ
­ÎÝÊÊ£x®Â
Divide both sides by 3.
JKL is equilateral.
4t - 8 = 2t + 1
2t = 9
4
18
The measure of each ∠ of
an equiangular is 60°.
B t
EXAMPLE
Example 4
Prove that the segment joining
the Mdpt. of two sides of an isosceles triangle is half the base.
Given: In isosceles ABC, X is
−−
the Mdpt. of AB, and Y is the
−−
Mdpt. of AC.
y
Using Coordinate Proof
Þ
Prove that the triangle whose vertices are the
midpoints of the sides of an isosceles triangle
is also isosceles.
−−
Given: ABC is isosceles. X is the mdpt. of AB.
−−
−−
Y is the mdpt. of AC. Z is the mdpt. of BC.
Prove: XYZ is isosceles.
­Ó>]ÊÓL®
8
8
äÊä®
Ý
­ä]Êä®
<
(
)
(
­{>]Êä®
)
2a + 4a _____
, 2b 2+ 0 = (3a, b), and the coordinates of Z
the coordinates of Y are ______
2
(
)
9
x
9
Proof:
Draw a diagram and place the coordinates of ABC and XYZ as shown.
2a + 0 _____
By the Midpoint Formula, the coordinates of X are _____
, 2b 2+ 0 = (a, b),
2
4a + 0 ____
are _____
, 0 +2 0 = (2a, 0).
2
Ó>ÊÓL®
2
2
By the Distance Formula, XZ = √(2a - a) + (0 - b) = √
a 2 + b 2 , and
(2a - 3a)2 + (0 - b)2 = √
YZ = √
a2 + b2.
{>Êä®
1
Prove: XY = __
AC.
2
By the Mdpt. Formula, the coords.
of X are (a, b), and Y are (3a, b).
By the Dist. Formula, XY = √
4a 2
= 2a, and AC = 4a. Therefore
1
XY = __
AC.
2
Also available on transparency
−− −−
Since XZ = YZ, XZ YZ by definition. So XYZ is isosceles.
4. What if...? The coordinates of ABC are A(0, 2b), B(-2a, 0),
and C(2a, 0). Prove XYZ is isosceles.
4- 8 Isosceles and Equilateral Triangles
Summarize
Remind students that an isosceles triangle
has at least two congruent sides, and its
properties can be used to prove that it
also has at least two congruent angles.
Emphasize that an equilateral triangle has
three congruent sides and angles. Review
the best methods of naming the vertices of
an isosceles and equilateral triangle in the
coordinate plane.
Questioning Strategies
E X A M P LE
3
• If the triangle is equiangular, how
do you find the measure of one of
its angles?
3 Close
ge07se_c04_0273_0279.indd 275
275
INTERVENTION
and INTERVENTION
Diagnose Before the Lesson
4-8 Warm Up, TE p. 273
Monitor During the Lesson
Check It Out! Exercises, SE pp. 274–275
Questioning Strategies, TE pp. 274–275
Assess After the Lesson
4-8 Lesson Quiz, TE p. 279
Alternative Assessment, TE p. 279
12/9/05 12:44:28 PM
E X A M P LE
4
• Why is it not suggested that you
use numerical values for the vertices in a coordinate proof?
Multiple Representations
For Example 4, you could
also place the triangle’s
vertices at (2a, 0), (-2a, 0), and
(0, 2b).
Lesson 4-8
275
Answers to Think and Discuss
THINK AND DISCUSS
1. An equil. is also an equiangular , so the 3 have the same
measure. They must add up to
180° by the Sum Thm. So
each ∠ must measure 60°.
1. Explain why each of the angles in an equilateral triangle measures 60°.
2. GET ORGANIZED Copy and complete the
/Àˆ>˜}i
graphic organizer. In each box, draw and
mark a diagram for each type of triangle.
µÕˆ>ÌiÀ>
µÕˆ>˜}Տ>À
2. See p. A4.
4-8
4-8 Exercises
Exercises
KEYWORD: MG7 4-8
KEYWORD: MG7 Parent
GUIDED PRACTICE
Assignment Guide
1. Vocabulary Draw isosceles JKL with ∠K as the vertex angle. Name the legs, base,
and base angles of the triangle.
Assign Guided Practice exercises
as necessary.
SEE EXAMPLE
If you finished Examples 1–2
Basic 12–16, 22–25, 28, 29,
32–34
Average 12–16, 22–25, 28, 29,
32–39
Advanced 12–16, 22–25, 28, 29,
32–39, 41, 45
2. Surveying To find the distance QR across a river, a surveyor locates three points Q,
R, and S. QS = 41 m, and m∠S = 35°. The measure of exterior ∠PQS = 70°. Draw a
diagram and explain how you can find QR.
1
p. 274
SEE EXAMPLE
Find each angle measure.
2
p. 274
3. m∠ECD
5. m∠X
If you finished Examples 1–4
Basic 12–25, 28–30, 42–44,
48–54
Average 12–23, 26–31, 33, 35,
36, 40, 42–44, 47–54
Advanced 12–21, 26–33, 35–54
6. m∠A
8
90°
9
­xÌÊʣήÂ
nÓÂ
<
27°
SEE EXAMPLE
49°
ΣÂ
4. m∠K
118°
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ÓÝÂ
p. 275
7. y
,
5
8. x
-
£ÓÞÂ
4
Homework Quick Check
Quickly check key concepts.
Exercises: 12, 14, 18, 21, 28, 30
­£äÝÊÊÓä®Â
/
9. BC
10. JK
20
Algebra In Exercise 11,
show students that
√
a 2 + a 2 ≠ a + a, or 2a.
Find each value.
3
ÞÊÊÓÎ
ÈÞÊÊÓ
p. 275
ÇÌÊÊ£x
£äÌ
11. Given: ABC is right isosceles. X is the
−− −− −−
midpoint of AC. AB BC
SEE EXAMPLE 4
50
­ä]ÊÓ>®
Prove: AXB is isosceles.
Þ
8
Ý
­ä]Êä®
276
Chapter 4 Triangle Congruence
Answers
1.
−−
−−
legs: KJ and KL
−−
base: JL
base: : ∠ J; ∠L
K
ge07se_c04_0273_0279.indd 276
J
L
2. By the Ext. ∠ Thm., m∠R = 35°. Since
m∠R = m∠S by the Conv. of the Isosc.
Thm., QR = QS = 41 m.
P
ƒ
KEYWORD: MG7 Resources
276
Chapter 4
­Ó>]Êä®
M
ƒ
Q
R
S
−−
11. It is given that ABC is rt. isosc., AB
−−
−−
BC, and X is the mdpt. of AC. By the
Mdpt. Formula, the coords. of X are
(a, a). By the Dist. Formula, AX = BX =
a √
2 . So AXB is isosc. by def. of an
isosc. .
12/9/05 12:44:30 PM
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
12
13–16
17–20
21
1
2
3
4
""
12. Aviation A plane is flying parallel
. When the
to the ground along AC
plane is at A, an air-traffic controller
in tower T measures the angle to
the plane as 40°. After the plane has
traveled 2.4 mi to B, the angle to the
plane is 80°. How can you find BT?
Ó°{ʓˆ
In Exercises 5, 6, 9, and 10, students may forget to substitute the
value of the variable back in the
equation to determine the answer to
the problem. Have them check their
answers as a way to avoid this.
nä¨
Extra Practice
{ä¨
Skills Practice p. S11
Ê,,",
,/
/
Application Practice p. S31
Find each angle measure.
13. m∠E
Answers
69°
,
14. m∠TRU
12. By the ∠ Add. Post., m∠ ATB =
40°. m∠BAT = 40° by the Alt. Int.
Thm. ∠ ATB ∠BAT by def. of
. Since ABT is isosc. by the
Converse of the Isosc. Thm.,
BT = BA = 2.4 mi.
−−
21. It is given that ABC is isosc. AB
−−
−−
AC, P is the mdpt. of AB, and
−−
Q is the mdpt. of AC. By the
Mdpt. Formula, the coords. of P
are (a, b), and the coords. of Q
are (3a, b) By the Dist. Formula,
−−
PC = QB = √
9a 2 + b 2 , so PC
−−
QB by the def. of .
33°
™ÈÂ
-
xÇÂ
1
/
15. m∠F
ÊÊÝÊÓÊ
ÊÂ
130° or 172° 16. m∠A
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31°
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Find each value.
17. z
92
18. y
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48
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21. Given: ABC is isosceles. P is the midpoint
−−
−−
of AB. Q is the midpoint of AC.
−− −−
AB AC
−− −−
Prove: PC QB
Þ
*
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+
Ý
­ä]Êä®
­{>]Êä®
Tell whether each statement is sometimes, always, or never true.
Support your answer with a sketch.
22. An equilateral triangle is an isosceles triangle. A
23. The vertex angle of an isosceles triangle is congruent to the base angles. S
24. An isosceles triangle is a right triangle. S
25. An equilateral triangle and an obtuse triangle are congruent. N
26. Critical Thinking Can a base angle of an isosceles triangle be an obtuse angle?
Why or why not?
4-8 PRACTICE A
4-8 PRACTICE C
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4- 8 Isosceles and Equilateral Triangles
Answers
25. Possible answer:
22.
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other base ∠ would also have to be
obtuse since they are . The sum of
the measures of the of the cannot
be greater than 180°.
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Lesson 4-8
277
Exercise 27 involves
finding angle measures of an isosceles
triangle. This exercise prepares students for the Multi-Step Test Prep on
page 280.
27. This problem will prepare you for the Multi-Step Test
Prep on page 280.
The diagram shows the inside view of the support
−− −−
structure of the back of a doghouse. PQ PR,
−− −−
PS PT, m∠PST = 71°, and m∠QPS = m∠RPT = 18°.
a. Find m∠SPT. 38°
*
+
b. Find m∠PQR and m∠PRQ. 53°
,
/
-
Answers
−−
30. It is given that ABC is isosc. BA
−−
−−
BC , and X is the mdpt. of AC.
Assign the coords. A(0, 2a),
B(0, 0) and C(2a, 0). By the
Mdpt. Formula, the coords. of X
are (a, a). By the Dist. Formula,
AX = XB = XC = a √
2 . So AXB
CXB by SSS.
Multi-Step Find the measure of each numbered angle.
28.
Î xnÂ
29.
m∠1 = 127°;
m∠2 = 26.5°;
m∠3 = 53°
Ç{Â
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30. Write a coordinate proof.
Ó
Given: ∠B is a right angle in isosceles right ABC.
−− −− −−
X is the midpoint of AC. BA BC
Prove: AXB CXB
31. Check students’ drawings. The are approximately 34°, 34°, and
112°. The conjecture should be
that the is isosc. The conjecture is correct since there are 2
.
35.
m∠1 = 58°;
m∠2 = 64°;
m∠3 = 122°
Ó£
8
31. Estimation Draw the figure formed by (-2, 1), (5, 5), and (-1, -7). Estimate
the measure of each angle and make a conjecture about the classification of the
figure. Then use a protractor to measure each angle. Was your conjecture correct?
Why or why not?
D
32. How many different isosceles triangles have a perimeter of 18 and sides whose
lengths are natural numbers? Explain.
4 : 5, 5, 8; 6, 6, 6; 7, 7, 4; 8, 8, 2
Multi-Step Find the value of the variable in each diagram.
E
X
33.
F
1. DEF (Given)
2. Draw the bisector of ∠EDF so
−−
that it intersects EF at X. (Every
∠ has a unique bisector.)
3. ∠EDX ∠FDX (Def. of ∠
bisector)
−− −−
4. DX DX (Reflex. Prop. of )
5. ∠E ∠F (Given)
6. EDX FDX (AAS Steps 3,
5, 4)
−− −−
7. DE DF (CPCTC)
36a. ∠B ∠C
b. Isosc. Thm.
c. Trans. Prop. of 37. DEF with ∠D ∠E ∠F
−−
is given. Since ∠E ∠F, DE
−−
DF by the Conv. of the Isosc.
Thm. Similarly, since ∠D −− −−
∠F, EF DE. By the Trans. Prop.
−− −−
of , EF DF. Combining the −− −− −−
statements, DE DF EF, and
DEF is equil. by def.
34.
20
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35. Prove the Converse of the Isosceles Triangle Theorem.
36. Complete the proof of Corollary 4-8-3.
−− −− −−
Given: AB AC BC
Prove: ∠A ∠B ∠C
−− −−
Proof: Since AB AC, a. ? by the Isosceles Triangle Theorem.
−−−−
−− −−
Since AC BC, ∠A ∠B by b. ? . Therefore ∠A ∠C by c. ? .
−−−−
−−−−
By the Transitive Property of , ∠A ∠B ∠C.
Navigation
37. Prove Corollary 4-8-4.
The taffrail log is
dragged from the stern
of a vessel to measure
the speed or distance
traveled during a
voyage. The log consists
of a rotator, recording
device, and governor.
38. Navigation The captain of a ship traveling along AB
sights an island C at an angle of 45°. The captain measures
the distance the ship covers until it reaches B, where
the angle to the island is 90°. Explain how to find the
distance BC to the island.
{xÂ
40. Write About It Write the Isosceles Triangle Theorem and its converse
as a biconditional. Two sides of a are if and only if
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™äÂ
39. Given: ABC CBA
Prove: ABC is isosceles.
the opp. those sides are .
38. By the Ext. ∠ Thm., m∠C = 45°,
278
Chapter 4 Triangle Congruence
so ∠ A ∠C. BC = AB by the
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4-8 RETEACH
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Conv. of the Isosc. Thm. So
the distance to island C is the
same as the distance traveled
from A to B.
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41. Rewrite the paragraph proof of the
Hypotenuse-Leg (HL) Congruence
Theorem as a two-column proof.
""
Given: ABC and DEF are right triangles.
∠C and ∠F are right angles.
−− −−
−− −−
AC DF, and AB DE.
Prove: ABC DEF
−− −−
Proof: On DEF draw EF
. Mark G so that FG = CB. Thus FG CB. From the diagram,
−− −−
−− −−
AC DF and ∠C and ∠F are right angles. DF ⊥ EG by definition of perpendicular
lines. Thus ∠DFG is a right angle, and ∠DFG ∠C. ABC DGF by SAS.
−−− −−
−−− −−
−− −−
DG AB by CPCTC. AB DE as given. DG DE by the Transitive Property.
By the Isosceles Triangle Theorem ∠G ∠E. ∠DFG ∠DFE since right angles
are congruent. So DGF DEF by AAS. Therefore ABC DEF by the
Transitive Property.
42. Lorena is designing a window so that ∠R, ∠S, ∠T, and
−− −−
∠U are right angles, VU VT, and m∠UVT = 20°.
What is m∠RUV?
10°
20°
70°
80°
43. Which of these values of y makes ABC isosceles?
1
1
1_
7_
4
2
1
1
2_
15_
2
2
ÓäÂ
-
In Exercise 34, students may write
the equation 5x + 15 = 60.
Encourage them to label all angle
measures before writing an equation.
If students have difficulty with Exercise
43, remind them that
they cannot assume which two sides
are congruent, so they must try all
possibilities.
/
Journal
6
Compare the hypothesis and the
conclusion of the Isosceles Triangle
Theorem with its converse. Support
your comparison with a sketch.
1
,
{Þ
ÎÞÊÊx
ÞÊÊ£ä
44. Gridded Response The vertex angle of an isosceles
triangle measures (6t - 9)°, and one of the base angles
measures (4t)°. Find t. 13.5
CHALLENGE AND EXTEND
Have students create a poster to
present examples of how to find
missing parts of an isosceles and
an equilateral triangle, when given
the measure of one angle or an
algebraic expression.
−− −−
−−− −−
45. In the figure, JK JL, and KM KL. Let m∠J = x°.
Prove m∠MKL must also be x°.
ÝÂ
46. An equilateral ABC is placed on a coordinate plane.
Each side length measures 2a. B is at the origin, and
C is at (2a, 0). Find the coordinates of A. a, a √
3
(
Ê,,",
,/
)
4-8
47. An isosceles triangle has coordinates A(0, 0) and B(a, b).
What are all possible coordinates of the third vertex?
Find each angle measure.
−−
(2a, 0), (0, 2b), or any pt. on the ⊥ bisector of AB
*
SPIRAL REVIEW
s
Y
48. x + 5x + 4 = 0
2
3 or 1
1
Find the slope of the line that passes through each pair of points. (Lesson 3-5)
(2, -1) and (0, 5)
51.
X
-3
52.
0
1. m∠R
_
(-5, -10) and (20, -10) 53. (4, 7) and (10, 11) 2
54. Position a square with a perimeter of 4s in the coordinate plane and give the
coordinates of each vertex. (Lesson 4-7)
s
­xÝÊʣǮ ­ÓÝÊÊ£ä®Â
50. x - 2x + 1 = 0
2
-4 or -1
ss
49. x - 4x + 3 = 0
2
+
,
Find the solutions for each equation. (Previous course)
54. Possible
answer:
28°
2. m∠P
124°
Find each value.
3
3. x
20
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4- 8 Isosceles and Equilateral Triangles
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−− −− −− −−
45. It is given that JK JL, KM KL, and
m∠ J = x°. By the Sum Thm.,
m∠ JKL + m∠JLK + x° = 180°. By the
Isosc. Thm., m∠ JKL = m∠ JLK.
So 2(m∠ JLK) + x° = 180° or m∠ JLK =
180 - x
m∠KML = m∠ JLK,
(______
)°. Since
2
180 - x
______
m∠KML = ( 2 )° by the Isosc. = 180° -
180 - x
180 - x
(______
)° - (______
)°.
2
2
Simplifying gives m∠MKL = x°.
ÚÓÊÞÊ ÊÊÎ
Î
41. See p. A17.
Thm. By the Sum Thm., m∠MKL +
m∠ JLK + m∠KML = 180° or m∠MKL
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12/9/05 12:44:40 PM
5. x
26°
ÝÂ
xÓÂ
6. The vertex angle of an isosceles
triangle measures (a + 15)°,
and one of the base angles
measures 7a°. Find a and
each angle measure.
a = 11; 26°; 77°; 77°
Also available on transparency
M AY BE C ONGR U ENT
C ANNOT BE C ONGR U ENT
—
X—
Lesson 4-8
279
SECTION 4B
SECTION
4B
Proving Triangles Congruent
Gone to the Dogs You are planning to build
a doghouse for your dog. The pitched roof of
the doghouse will be supported by four trusses.
Each truss will be an isosceles triangle with the
dimensions shown. To determine the materials
you need to purchase and how you will construct
the trusses, you must first plan carefully.
Organizer
GI
Objective: Assess students’
ability to apply concepts and skills
in Lessons 4-4 through Lesson 4-8
in a real-world format.
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@<I
™Êˆ˜°
Online Edition
Ó{ʈ˜°
−− −−
1. Measure
AB, BC,1.
−−
and CA. If these
3 lengths are the
same for every truss,
then the trusses all
have the same size
and shape by SSS.
2.
Resources
Geometry Assessments
www.mathtekstoolkit.org
Problem
Text Reference
1
Lesson 4-4
2
Lesson 4-5
3
Lesson 4-6
4
Lesson 4-7
5
Lesson 4-8
6
Lessons 4-4 to 4-8
You want to be sure that all four
trusses are exactly the same size
and shape. Explain how you
could measure three lengths
on each truss to ensure this.
Which postulate or theorem
are you using?
Prove that the two triangular
halves of the truss are congruent.
−− −−
−−
3. What can you say about AD
3. AD DB by
−−
and DB? Why is this true?
CPCTC. AD = DB
Use this to help you find the
= 12 in. and
−−
−− −− −−
lengths of AD, DB, AC, and BC.
AC = BC = 15 in.
4. You want to make careful plans on a coordinate plane
4. Possible answer: before you begin your construction of the trusses.
Each unit of the coordinate plane represents 1 inch.
A(0, 0), B(24, 0),
How could you assign coordinates to vertices A, B, and C?
Answers
C(12, 9)
−−
−−
2. 1. CD ⊥ AB (Given)
2. ∠CDA and ∠CDB are rt. .
(Def. of ⊥)
3. CDA and CDB are rt. .
(Def. of rt. )
−− −−
4. AC BC (Given)
−− −−
5. CD CD (Reflex. Prop. of )
6. CDA CDB (HL Steps 4, 5)
5. m∠ACB = 106°. What is the measure of each of the acute 5. m∠A = m∠B = 37°;
angles in the truss? Explain how you found your answer. the base of an
isosc. are , so
6. You can buy the wood for the trusses at the building
supply store for $0.80 a foot. The store sells the wood in 2m∠A + 106° = 180°.
6-foot lengths only. How much will you have to spend
to get enough wood for the 4 trusses of the doghouse? $14.40
280
Chapter 4 Triangle Congruence
INTERVENTION
Scaffolding Questions
ge07se_c04_0280_0281.indd 280
1. Which postulate or theorem applies if
you know that three sides of one triangle
are congruent to three sides of another
triangle? SSS
KEYWORD: MG7 Resources
280
Chapter 4
2. What type of triangle is formed by each
half of the truss? rt.
−− −−
3. Can you conclude that AD DB ? If so,
why? Yes; CPCTC Once you know AD,
how can you find AC ? Pyth. Thm.
−−
4. If you place A at the origin and AB along
the x-axis, what are the coordinates of B?
Why? B(24, 0); AB = 24
5. What can you say about the acute angles
of the triangle? Why? They are ; they are
base of an isosc. .
6. Suppose that you can buy wooden
boards for $1.80 per foot. The store sells
the boards in whole feet only. How much
will you have to spend in order to buy
enough wood to make the trusses for the
doghouse? $9
Extension
Which congruence postulates or theorems
involve only one pair of congruent sides?
ASA; AAS Can they be applied to prove
ACD BCD? Yes
5/8/06 1:08:20 PM
SECTION 4B
SSEECCTTIIOONN
Quiz for Lessons 4-4 Through 4-8
4B
2A
4-4 Triangle Congruence: SSS and SAS
1. The figure shows one tower and the cables of a suspension bridge.
−− −−
Given that AC BC, use SAS to explain why ACD BCD.
−−
−− −−
2. Given: JK bisects ∠MJN. MJ NJ
Prove: MJK NJK
Organizer
Objective: Assess students’
mastery of concepts and skills in
Lessons 4-4 through 4-8.
4-5 Triangle Congruence: ASA, AAS, and HL
Determine if you can use the HL Congruence Theorem to prove the triangles
congruent. If not, tell what else you need to know.
3. RSU and TUS
,
-
1
/
−− −−
no; AC DB
yes
Resources
4. ABC and DCB
Assessment Resources
Observers in two lighthouses K and L spot a ship S.
5. Draw a diagram of the triangle formed by the
lighthouses and the ship. Label each measure.
6. Is there enough data in the table to pinpoint
the location of the ship? Why?
K to L
K to S
L to S
Bearing
E
N 58° E
N 77° W
Distance
12 km
?
?
5.
Yes; the is uniquely determined by ASA.
4-6 Triangle Congruence: CPCTC
S
INTERVENTION
ƒ
ƒ
K
−− −− −− −−
7. Given: CD BE, DE CB
Prove: ∠D ∠B
Section 4B Quiz
KM
L
Resources
Ready to Go On?
Intervention and
Enrichment Worksheets
Check students’ work; possible answer:
4-7 Introduction to Coordinate Proof vertices at (0, 0), (9, 0), (9, 9), and (0, 9).
8. Position a square with side lengths of 9 units in the coordinate plane
Ready to Go On? CD-ROM
9. Assign coordinates to each vertex and write a coordinate proof.
−−
−−
Given: ABCD is a rectangle with M as the midpoint of AB. N is the midpoint of AD.
1
Prove: The area of AMN is __8 the area of rectangle ABCD.
Ready to Go On? Online
4-8 Isosceles and Equilateral Triangles
Find each value.
10. m∠C
100°
Answers
,
11. ST
1, 2, 7, 9, 12. See p. A17.
6
ÓÝÂ
xÝÂ
nÊÊ{Ü
ÓÜÊÊx
/
-
12. Given: Isosceles JKL has coordinates J(0, 0), K (2a, 2b), and L(4a, 0).
−−
−−
M is the midpoint of JK, and N is the midpoint of KL.
Prove: KMN is isosceles.
Ready to Go On?
NO
ge07se_c04_0280_0281.indd 281
READY
Ready to Go On?
Intervention
YES
Diagnose and Prescribe
INTERVENE
TO
Worksheets
281
ENRICH
8/15/05 10:27:46 AM
GO ON? Intervention, Section 4B
CD-ROM
Lesson 4-4
4-4 Intervention
Activity 4-4
Lesson 4-5
4-5 Intervention
Activity 4-5
Lesson 4-6
4-6 Intervention
Activity 4-6
Lesson 4-7
4-7 Intervention
Activity 4-7
Lesson 4-8
4-8 Intervention
Activity 4-8
Online
Diagnose and
Prescribe Online
READY TO GO ON?
Enrichment, Section 4B
Worksheets
CD-ROM
Online
Ready to Go On?
281
CHAPTER
Organizer
4
EXTENSION
Proving Constructions Valid
Pacing: Traditional 1 day
Block
__1 day
2
Objective
Use congruent triangles
to prove constructions
valid.
Objective: Use congruent
GI
triangles to prove constructions
valid.
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When performing a compass and straight edge construction, the compass
setting remains the same width until you change it. This fact allows you to
construct a segment congruent to a given segment. You can assume that two
distances constructed with the same compass setting are congruent.
Online Edition
Using the Extension
The steps in the construction of a figure can be justified by combining the
assumptions of compass and straightedge constructions and the postulates and
theorems that are used for proving triangles congruent.
In Chapter 4, students learn to prove
triangles congruent and to use properties of congruent triangles in other
proofs. In this extension, students
use the properties of congruent
triangles to prove constructions valid.
You have learned that there exists exactly one midpoint on any line segment.
The proof below justifies the construction of a midpoint.
EXAMPLE
Proving the Construction of a Midpoint
Given: diagram showing the steps
in the construction
−−
Prove: M is the midpoint of AB .
Answers to Check It Out!
−− −− −−
−−
1. 1. Draw AC, BC, AD, and BD.
(Through any 2 pts. there is
exactly 1 line.)
−− −− −− −−
2. AC BC AD BD (Same
compass setting used)
−− −−
3. CD CD (Reflex. Prop. of )
4. ACD BCD (SSS Steps
2, 3)
5. ∠ ACD ∠BCD (CPCTC)
−− −−
6. CM CM (Reflex. Prop. of )
7. ACM BCM (SAS Steps
2, 5, 6)
8. ∠ AMC ∠BMC (CPCTC)
9. ∠ AMC and ∠BMC are rt. .
( supp. → rt. )
−−
−−
10. CD ⊥ AB (Def. of ⊥)
−− −−
11. AM BM (CPCTC)
−−

bisects AB. (Def. of
12. CD
bisector)
1
To construct a
midpoint, see the
construction of
a perpendicular
bisector on p. 172.
Proof:
Statements
−− −− −−−
−−
1. Draw AC, BC, AD, and BD .
−−
2. AC −−
3. CD −− −−− −−
BC AD BD
−−
CD
Reasons
1. Through any two pts. there is
exactly one line.
2. Same compass setting used
3. Reflex. Prop. of 4. ACD BCD
4. SSS Steps 2, 3
5. ∠ACD ∠BCD
−−− −−−
6. CM CM
5. CPCTC
7. ACM BCM
−−− −−−
8. AM BM
7. SAS Steps 2, 5, 6
−−
9. M is the midpt. of AB.
6. Reflex. Prop. of 8. CPCTC
9. Def. of mdpt.
1. Given: above diagram
−−
is the perpendicular bisector of AB.
Prove: CD
282
Chapter 4 Triangle Congruence
1 Introduce
ge07se_c04_0282_0283.indd 282
2 Teach
Motivate
Guided Instruction
Point out to students that some assumptions made about their construction tools,
such as a compass setting being fixed,
could make their constructions inaccurate.
Discuss what characteristics of their construction tools might make their constructions invalid. Possible answer: Compass
does not hold its setting; compass does
not hold the pencil firmly; straightedge is
cracked or broken.
Explain that steps in a construction contain given information much like the tick
marks in a diagram show that segments
are congruent. Have students practice identifying what information is given from the
construction marks in a diagram. Explain to
students that justifying their constructions
can provide a review of concepts from
previous lessons.
Inclusion Remind students to
make their constructions big
enough for easy use of their
KEYWORD: MG7 Resources
282
Chapter 4
compass.
10/26/05 6:08:34 PM
EXAMPLE
2
Proving the Construction of an Angle
Additional Examples
Given: diagram showing the steps in the construction
Prove: ∠A ∠D
Given: Diagram showing the
steps in the construction
−−
−−
Prove: CD ⊥ AB.
To review the
construction of an
angle congruent to
another angle, see
page 22.
Proof: Since there is a straight line through any two points, you can draw
−−
−− −− −−
−−
BC and EF. The same compass setting was used to construct AC, AB, DF,
−− −− −− −−
−−
and DE, so AC AB DF DE. The same compass setting was used
−−
−− −−
−−
to construct BC and EF, so BC EF. Therefore BAC EDF by SSS,
and ∠A ∠D by CPCTC.
2. Prove the construction for
bisecting an angle. (See page 23.)
−−
−−
Draw BD and CD (through any 2 pts. there is
exactly 1 line).
same−−
compass
−− the−−
−− Since
−− setting
−−
was used, AB AC and BD CD. AD AD by the
Reflex. Prop. of . So ABD ACD by SSS,

and ∠BAD ∠CAD by CPCTC. Therefore AD
bisects ∠BAC by the def. of ∠ bisector.
EXTENSION
Example 1
Exercises
−− −−
1. Draw AC, BC. (Through any two
pts. there is exactly one line.)
−− −−
2. AC BC (Same compass setting
used)
−− −−
3. AD BD (Same compass
setting used)
−− −−
4. CD CD (Reflex. Prop. of )
5. ADC BDC (SSS Steps 2,
3, 4)
6. ∠ ADC ∠BDC (CPCTC)
7. ∠ ADC and ∠BDC are rt. . ( that form a lin. pair are rt. .)
−−
−−
8. CD ⊥ AB (Def. of ⊥)
Example 2
Use each diagram to prove the construction valid.
1. parallel lines
(See page 163 and page 170.)
Given: Diagram showing the
steps in the construction
Prove: ∠D ∠A
2. a perpendicular through a point not
on the line (See page 179.)
*
+
3. constructing a triangle using SAS
(See page 243.)
4. constructing a triangle using ASA
(See page 253.)
Extension
283
Since there is a straight line
through any two points, you
−−
−−
can draw BC and EF. The same
compass setting was used to
−− −− −−
−−
construct AC, AB, DF, and DE,
−− −− −− −−
so AC AB DF DE. The
same compass setting was used
−−
−−
−−
to construct BC and EF, so BC
−−
EF. Therefore ABC DEF
by SSS, and ∠D ∠A by CPCTC.
Also available on transparency
3 Close
ge07se_c04_0282_0283.indd 283
Summarize
Review the steps for proving a construction
valid.
• Identify congruent segments constructed
with the same compass setting.
• Use triangle congruence and CPCTC (or
other theorems and postulates) to complete the proof.
Answers
−−
−−
1. Draw auxiliary segments BC and EF.
(Through any 2 pts. there is exactly 1
line.) Since the same compass setting
−−
−− −− −− −−
was used, AB AC DE DF and BC
−−
EF. BAC EDF by SSS. ∠BAC


∠EDF by CPCTC. Therefore DF
AC
by the Conv. of the Corr. Thm.
2–4. See p. A17.
INTERVENTION
10/26/05 6:08:35 PM
Questioning Strategies
E X A M P LES 1 – 2
• Why is it necessary to first prove
congruent triangles when proving
constructions valid?
Extension
283
CHAPTER
For a complete
list of the
postulates and
theorems in
this chapter,
see p. S82.
Study Guide:
Review
4
Organizer
Vocabulary
CPCTC . . . . . . . . . . . . . . . . . . . . . 260
isosceles triangle . . . . . . . . . . . 217
auxiliary line . . . . . . . . . . . . . . . 223
equiangular triangle . . . . . . . . 216
legs of an isosceles triangle . . 273
organize and review key concepts
and skills presented in Chapter 4.
base . . . . . . . . . . . . . . . . . . . . . . . 273
equilateral triangle . . . . . . . . . 217
obtuse triangle . . . . . . . . . . . . . 216
base angle . . . . . . . . . . . . . . . . . . 273
exterior . . . . . . . . . . . . . . . . . . . . 225
remote interior angle . . . . . . . 225
congruent polygons . . . . . . . . . 231
exterior angle . . . . . . . . . . . . . . 225
right triangle . . . . . . . . . . . . . . . 216
coordinate proof . . . . . . . . . . . . 267
included angle. . . . . . . . . . . . . . 242
scalene triangle . . . . . . . . . . . . . 217
corollary . . . . . . . . . . . . . . . . . . . 224
included side . . . . . . . . . . . . . . . 252
triangle rigidity . . . . . . . . . . . . . 242
corresponding angles . . . . . . . 231
interior . . . . . . . . . . . . . . . . . . . . 225
vertex angle . . . . . . . . . . . . . . . . 273
corresponding sides. . . . . . . . . 231
interior angle . . . . . . . . . . . . . . . 225
GI
acute triangle . . . . . . . . . . . . . . 216
Objective: Help students
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Online Edition
Multilingual Glossary
Countdown to
Testing Week 9
Complete the sentences below with vocabulary words from the list above.
1. A(n) ? is a triangle with at least two congruent sides.
−−−−
2. A name given to matching angles of congruent triangles is
Resources
3. A(n)
?
−−−−
? .
−−−−
is the common side of two consecutive angles in a polygon.
Multilingual Glossary Online
4-1 Classifying Triangles (pp. 216–221)
KEYWORD: MG7 Glossary
EXERCISES
EXAMPLE
■
Lesson Tutorial Videos
CD-ROM
Classify the triangle by its angle measures
and side lengths.
isosceles right triangle
Classify each triangle by its angle measures and
side lengths.
4.
5.
ÈäÂ
ÈäÂ
ÈäÂ
£ÎxÂ
Answers
1. isosceles triangle
4-2 Angle Relationships in Triangles (pp. 223–230)
2. corresponding angles
3. included side
EXERCISES
EXAMPLE
4. equiangular; equil.
■
12x = 3x + 42 + 6x
12x = 9x + 42
Find m∠S.
5. obtuse; scalene
­ÎÝÊ{Ӯ /
6. 60°
£ÓÝÂ
,
7. 66.5°
ÈÝÂ
-
Find m∠N.
6.
ÞÂ
3x = 42
x = 14
m∠S = 6 (14) = 84°
*
ÞÂ
£ÓäÂ
+
7. InLMN, m∠L = 8x °, m∠M = (2x + 1)°, and
m∠N = (6x - 1)°.
284
Chapter 4 Triangle Congruence
ge07se_c04_0284_0293.indd 284
284
Chapter 4
12/3/05 7:02:21 PM
−−
8. XZ
4-3 Congruent Triangles (pp. 231–237)
EXERCISES
EXAMPLE
■
9. ∠Q
10. 25
Given: DEF JKL. Identify all pairs of
congruent corresponding parts.
Then find the value of x.
Given: ABC CDA
Find each value.
­nÝÊÊÓÓ®Â
10. x
−− −− −− −−
12. 1. AB DE, DB AE (Given)
−− −−
2. AD DA (Reflex. Prop. of )
3. ADB DAE (SSS Steps
1, 2)
−−
−−
−−
13. 1. GJ bisects FH, and FH
−−
bisects GJ. (Given)
−− −− −− −−
2. GK JK, FK HK (Def. of seg.
bisect)
{ÇÂ
£xÊÊ{Þ
ÎÞÊ£
11. CD
11. 7
Given: PQR XYZ. Identify the congruent
corresponding parts.
−−
8. PR ?
9. ∠Y ?
−−−−
−−−
The congruent pairs follow: ∠D ∠J, ∠E ∠K,
−− −− −− −−
−− −−
∠F ∠L, DE JK, EF KL, and DF JL.
­ÓÝÊÊήÂ
3. ∠GKF ∠JKH (Vert. Thm.)
Since m∠E = m∠K, 90 = 8x - 22. After 22 is
added to both sides, 112 = 8x. So x = 14.
4. FGK HJK (SAS Steps
2, 3)
14. BC = (-6)2+ 36 = 72; YZ =
−− −−
2(-6)2 = 72; BC YZ; ∠C −− −−
∠Z; AC XZ. So ABC XYZ
by SAS.
4-4 Triangle Congruence: SSS and SAS (pp. 242–249)
−− −−
■ Given: RS UT, and
−− −−
VS VT. V is
the midpoint
−−
of RU.
,
6
Statements
−− −−
1. RS UT
−− −−
2. VS VT
−− −−
12. Given: AB DE,
−− −−
DB AE
Prove: ADB DAE
/
-
Prove: RSV UTV
Proof:
Reasons
1
−−
−−
13. Given: GJ bisects FH,
−−
−−
and FH bisects GJ.
Prove: FGK HJK
2. Given
3. Given
14. Show that ABC XYZ when x = -6.
5. SSS Steps 1, 2, 4
ÓÝ Ó
Ý ÓÊÊÎÈ
Show that ADB CDB when s = 5.
£{ÊÓÃ
{
{ÓÂ
8
{ÓÂ
nÓ
<
nÓ
15. Show that LMN PQR when y = 25.
x
ÃÊÓ
9
4. Def. of mdpt.
5. RSV UTV
à ÓÊ{Ã
1. Given
−−
3. V is the mdpt. of RU.
−− −−
4. RV UV
■
15. PQ = 25 - 1 = 24; QR = 25;
and PR = 25 2 - (25 - 1)2 - 42
−− −− −− −− −−
= 7; LM PQ; MN QR; LN
−−
PR; so LMN PQR by
SSS.
EXERCISES
EXAMPLES
AB = s 2 - 4s
AD = 14 - 2s
= 5 2 - 4 (5 )
= 14 - 2 (5 )
=5
=4
−− −−
−− −−
BD BD by the Reflexive Property. AD CD
−− −−
and AB CB. So ADB CDB by SSS.
Ç
Óx
Ó{
,
ÞÓÊÊ­ÞÊ£®ÓÊÊ{Ó
*
Þ
ÞÊ£
+
Study Guide: Review
ge07se_c04_0284_0293.indd 285
285
12/3/05 7:02:29 PM
Study Guide: Review
285
Answers
4-5 Triangle Congruence: ASA, AAS, and HL (pp. 252–259)
−−
16. 1. C is the mdpt. of AG. (Given)
−− −−
2. GC AC (Def. of mdpt.)
−− −−
3. HA GB (Given)
4. ∠HAC ∠BGC (Alt. Int. Thm.)
5. ∠HCA ∠BCG (Vert. Thm.)
6. HAC BGC (ASA Steps 4,
2, 5)
−−
−− −−
−−
17. 1. WX ⊥ XZ, YZ ⊥ ZX (Given)
2. ∠WXZ and ∠YZX are rt. .
(Def. of ⊥)
3. WZX and YXZ are rt. .
(Def. of rt. )
−− −−
4. XZ XZ (Reflex. Prop. of )
−− −−
5. WZ YX (Given)
6. WZX YXZ (HL Steps
5, 4)
18. 1. ∠S and ∠V are rt. . (Given)
2. ∠S ∠V (Rt. ∠ Thm.)
3. RT = UW (Given)
−− −−
4. RT UW (Def. of )
5. m∠T = m∠W (Given)
6. ∠T ∠W (Def. of )
7. RST UVW (AAS Steps 2,
6, 4)
−−
19. 1. M is the mdpt. of BD. (Given)
−− −−
2. MB DM (Def. of mdpt.)
−− −−
3. BC DC (Given)
−− −−
4. CM CM (Reflex. Prop. of )
5. CBM CDM (SSS Steps
2, 3, 4)
6. ∠1 ∠2 (CPCTC)
−− −−
20. 1. PQ RQ (Given)
−− −−
2. PS RS (Given)
−− −−
3. QS QS (Reflex. Prop. of )
4. PQS RQS (SSS Steps 1,
2, 3)
5. ∠PQS ∠RQS (CPCTC)
−−
6. QS bisects ∠PQR. (Def. of
bisect)
−−
21. 1. H is mdpt. of GJ, and L is
−−
mdpt. of MK. (Given)
2. GH = JH, ML = KL (Def. of
mdpt.)
−− −− −− −−
3. GH JH, ML KL (Def. of )
−− −−
4. GJ KM (Given)
−− −−
5. GH KL (Div. Prop. of )
−− −−
6. GM KJ, ∠G ∠K (Given)
7. GMH KJL (SAS Steps 5,
6)
8. ∠GMH ∠KJL (CPCTC)
−−
■ Given: B is the midpoint of AE.
∠A ∠E,
∠ABC ∠EBD
Prove: ABC EBD
Proof:
Statements
Reasons
1. ∠A ∠E
1. Given
2. ∠ABC ∠EBD
2. Given
−−
3. B is the mdpt. of AE.
−− −−
4. AB EB
5. ABC EBD
−−− −−
17. Given: WX ⊥ XZ,
−− −−
YZ ⊥ ZX,
−−− −−
WZ YX
Prove: WZX YXZ
3. Given
9
8
<
7
4. Def. of mdpt.
5. ASA Steps 1, 4, 2
18. Given: ∠S and ∠V
are right angles.
RT = UW.
m∠T = m∠W
Prove: RST UVW
,
/
-
1
6
7
4-6 Triangle Congruence: CPCTC (pp. 260–265)
EXERCISES
EXAMPLES
−−
−−
■ Given: JL and HK bisect each other.
Prove: ∠JHG ∠LKG
Proof:
Statements
−−
−−
1. JL and HK bisect
each other.
−− −−
2. JG LG, and
−−− −−
HG KG.
Reasons
1. Given
19. Given: M is the midpoint
−−
of BD.
−− −−
BC DC
Prove: ∠1 ∠2
£
Ó
−− −−
20. Given: PQ RQ,
−− −−
PS RS
−−
Prove: QS bisects ∠PQR.
3. Vert. Thm.
4. JHG LKG
4. SAS Steps 2, 3
5. ∠JHG ∠LKG
5. CPCTC
+
-
2. Def. of bisect
3. ∠JGH ∠LGK
−−
21. Given: H is the midpoint of GJ.
−−−
L is the midpoint of MK.
−−− −− −− −−−
GM KJ, GJ KM ,
∠G ∠K
Prove: ∠GMH ∠KJL
,
*
Chapter 4 Triangle Congruence
ge07se_c04_0284_0293.indd 286
Chapter 4
16. Given: C is the midpoint
−−
of AG.
−− −−
HA GB
Prove: HAC BGC
286
286
EXERCISES
EXAMPLES
5/8/06 1:09:12 PM
Answers
4-7 Introduction to Coordinate Proof (pp. 267–272)
Given: ∠B is a right angle in isosceles right
−−
ABC. E is the midpoint of AB.
−− −− −−
D is the midpoint of CB. AB CB
−− −−
Prove: CE AD
Proof: Use the coordinates A(0, 2a) , B(0, 0),
−−
−−
and C(2a, 0). Draw AD and CE.
y
s
EXERCISES
EXAMPLES
■
22. (0, 0), (r, 0), (0, s)
Þ
Ý
By the Midpoint Formula,
0 + 0 2a + 0
E = _, _ = (0, a) and
2
2
0 + 2a 0 + 0
D = _, _ = (a, 0)
2
2
By the Distance Formula,
(2a - 0)2 + (0 - a)2
CE = √
(
(
)
)
Position each figure in the coordinate plane and give
the coordinates of each vertex.
22. a right triangle with leg lengths r and s
x
23. a rectangle with length 2p and width p
23. (0, 0), (2p, 0), (2p, p), (0, p)
24. a square with side length 8m
y
For exercises 25 and 26 assign coordinates to each
vertex and write a coordinate proof.
25. Given: In rectangle ABCD, E is the midpoint of
−−
−−
AB, F is the midpoint of BC, G is the
−−
midpoint of CD, and H is the midpoint
−−
of AD.
−− −−−
Prove: EF GH
p
pp
p
x
24. (0, 0), (8m, 0), (8m, 8m),
(0, 8m)
y
m
26. Given: PQR has a right ∠Q .
−−
M is the midpoint of PR .
Prove: MP = MQ = MR
= √
a 2 + 4a 2 = a √
5
−− −−
Thus CE AD by the definition of congruence.
=
26. Use coords. P(0, 2b), Q(0, 0),
and R(2a, 0). Then by the Mdpt.
Formula, the cords. are M(a, b).
By the Dist. Formula,
(a - 0) 2 + (b - 0) 2
QM = √
EXERCISES
6x = 138
x = 23
values.
Simplify.
Find each value.
28. x
­{xÊÊÎÝ®Â
=
,
ÓÞÊÊ{°x
/
√
a2 + b2,
(a - 0) 2 + (b - 2b) 2
PM = √
a 2 + b 2 , and
= √
29. RS
√
a 2 + b 2 , and
(0 - a) 2 + (b - 2b) 2
GH = √
−−
−−
2
+ b 2 . So EF GH by the
= √a
def. of .
4-8 Isosceles and Equilateral Triangles (pp. 273–279)
Find the value of x.
ÎÝÂ
m∠D + m∠E + m∠F = 180°
by the Triangle Sum
{ÓÂ
Theorem. m∠E = m∠F
by the Isosceles
Triangle Theorem.
m∠D + 2 m∠E = 180° Substitution
42 + 2 (3x) = 180
Substitute the given
m
25. Use coords. A(0, 0), B(2a, 0),
C(2a, 2b), and D(0, 2b). Then by
the Mdpt. Formula, the cords. are
E(a, 0), F(2a, b), G(a, 2b), and
H(0, b). By the Dist. Formula,
(2a - a) 2 + (b - 0) 2
EF = √
(a - 0)2 + (0 - 2a)2
AD = √
■
mm
x
27. Show that a triangle with vertices at (3, 5), (3, 2),
and (2, 5) is a right triangle.
= √
4a 2 + a 2 = a √
5
EXAMPLE
r
£°xÞ
(2a - a) 2 + (0 - b) 2
RM = √
2
+ b 2 . So QM = PM = RM.
= √a
By def. M is equidistant from the
vertices of PQR.
-
Divide both sides by 6.
30. Given: ACD is isosceles with ∠D as the vertex
−−
angle. B is the midpoint of AC .
AB = x + 5, BC = 2x - 3, and CD = 2x + 6.
Find the perimeter of ACD.
Study Guide: Review
287
27. In a rt. , a 2 + b 2 = c 2.
(3 - 3) 2 + (5 - 2) 2 = 3,
√
(3 - 2) 2 + (2 - 5) 2 = √
10 ,
√
2
2
√(2 - 3) + (5 - 5) = 1, and
10 ) .
3 2 + 1 2 = ( √
Since 9 + 1 = 10, it is a rt. .
2
28. -5
ge07se_c04_0284_0293.indd 287
12/3/05 7:02:47 PM
29. 13.5
30. 70 units
Study Guide: Review
287
CHAPTER
4
Organizer
1. Classify ACD by its angle measures. rt.
2. ACD
GI
mastery of concepts and skills in
Chapter 4.
<
x°Ç
Classify each triangle by its side lengths.
Objective: Assess students’
D@<I
3. ABC
scalene
4. ABD
isosc.
Given: XYZ JKL
Identify the congruent corresponding parts.
−−
−−
6. JL ?
7. ∠Y ?
XZ
∠K
−−−−
−−−−
−−
−−
10. Given: T is the midpoint of PR and SQ.
Prove: PTS RTQ
Resources
Assessment Resources
Chapter 4 Tests
• Free Response
(Levels A, B, C)
Î
scalene
,
5. While surveying the triangular plot of land shown,
a surveyor finds that m∠S = 43°. The measure
of ∠RTP is twice that of ∠RTS. What is m∠R? 77°
Online Edition
x
{ÎÂ
-
/
8. ∠L *
−−
9. YZ ?
∠Z
−−−−
−−
?
KL
−−−−
*
/
-
+
,
• Multiple Choice
(Levels A, B, C)
11. The figure represents a walkway with
−−
triangular supports. Given that GJ bisects
∠HGK and ∠H ∠K, use AAS to prove
HGJ KGJ
• Performance Assessment
IDEA Works! CD-ROM
Modified Chapter 4 Test
−− −−
12. Given: AB DC,
−− −−
AB ⊥ AC,
−− −−
DC ⊥ DB
Prove: ABC DCB
−− −−
13. Given: PQ SR,
∠S ∠Q
−− −−
Prove: PS QR
*
+
-
,
14. Position a right triangle with legs 3 m and 4 m long in the coordinate plane.
Give the coordinates of each vertex.
Answers
−−
−−
10. 1. T is the mdpt. of PR and SQ.
(Given)
−− −− −− −−
2. PT RT, ST QT (Def. of
mdpt.)
3. ∠PTS ∠RTQ (Vert. Thm.)
4. PTS RTQ (SAS Steps
2, 3)
15. Assign coordinates to each vertex and write a coordinate proof.
Given: Square ABCD
−− −−
Prove: AC BD
Find each value.
16. y
17. m∠S
-5
­xÊÊ££Þ®Â
-
*
44°
xÈÂ
,
/
+
18. Given: Isosceles ABC has coordinates A(2a, 0), B(0, 2b), and C(-2a, 0).
−−
−−
D is the midpoint of AC, and E is the midpoint of AB.
Prove: AED is isosceles.
288
Chapter 4 Triangle Congruence
Answers
11. 1. ∠H ∠K (Given)
−−
2. GJ bisects ∠HGK. (Given)
3. ∠ HGJ ∠ KGJ (Def. of bisect)
−− −−
4. JG JG (Reflex. Prop. of )
5. HGJ KGJ (AAS Steps 1, 3, 4)
−− −−
−−
−−
12. 1. AB ⊥ AC, DC ⊥ DB (Given)
2. ∠BAC and ∠CDB are rt. . (Def. of
⊥)
3. ABC and DCB are rt. . (Def. of
rt. )
−− −−
4. AB DC (Given)
−− −−
5. BC CB (Reflex. Prop. of )
6. ABC DCB (HL Steps 5, 4)
ge07se_c04_0284_0293.indd 288
KEYWORD: MG7 Resources
288
Chapter 4
−− −−
13. 1. PQ SR (Given)
2. ∠QPR ∠SRP (Alt. Int. Thm.)
3. ∠S ∠Q (Given)
−− −−
4. PR RP (Reflex. Prop. of )
5. QPR SRP (AAS Steps 2, 3, 4)
6. ∠SPR ∠QRP (CPCTC)
−− −−
7. PS QR (Conv. of Alt. Int. Thm.)
y
m
14.
x
m
15, 18. See p. A17.
6/10/06 1:38:53 PM
CHAPTER
4
Organizer
FOCUS ON ACT
The ACT Mathematics Test is one of four tests in the
ACT. You are given 60 minutes to answer 60 multiplechoice questions. The questions cover material typically
taught through the end of eleventh grade. You will need
to know basic formulas but nothing too difficult.
Objective: Provide practice for
There is no penalty for guessing on the
ACT. If you are unsure of the correct
answer, eliminate as many answer choices
as possible and make your best guess.
Make sure you have entered an answer
for every question before time runs out.
GI
college entrance exams such as the
ACT.
<D
@<I
Online Edition
You may want to time yourself as you take this practice test.
It should take you about 5 minutes to complete.
1. For the figure below, which of the following
must be true?
Resources
3. Which of the following best describes a triangle
with vertices having coordinates (-1, 0), (0, 3),
and (1, -4)?
College Entrance Exam
Practice
(A) Equilateral
Questions on the ACT represent the
following content areas:
(B) Isosceles
(C) Right
Pre-Algebra, 23%
I. m∠EFG > m∠DEF
(D) Scalene
Elementary Algebra, 17%
II. m∠EDF = m∠EFD
(E) Equiangular
Intermediate Algebra, 15%
III. m∠DEF + m∠EDF > m∠EFG
(A) I only
Coordinate Geometry, 15%
Plane Geometry, 23%
4. In the figure below, what is the value of y?
(B) II only
Trigonometry, 7%
ÞÂ
(C) I and II only
£ÎÈÂ
(D) II and III only
Items on this page focus on:
£Î£Â
• Elementary Algebra
(F) 49
(E) I, II, and III
• Coordinate Geometry
(G) 87
2. In the figure below, ABD CDB,
m∠A = (2x + 14)°, m∠C = (3x - 15)°, and
m∠DBA = 49°. What is the measure of ∠BDA?
• Plane Geometry
(H) 93
(J) 131
Text References:
(K) 136
Item
Lesson
(F) 29°
(G) 49°
(H) 59°
(J) 72°
(K) 101°
1. Remind students that the sum of the
angle measures of a triangle is 180° and
that the measure of an exterior angle
is equal to the sum of the measures of
its remote interior angles. These facts,
along with the diagram, lead to the conclusion that statements I and II are true
but III is false.
2. Students may choose G because they
mislabeled the figure or misinterpreted
the names of the angles. Students may
choose F because they only solved for x.
Remind students to read each test item
completely.
2
3
4
5
4-2
4-3
4-7
4-2
4-8
5. In RST, RS = 2x + 10, ST = 3x - 2, and
RT = __12 x + 28. If RST is equiangular, what
is the value of x?
(A) 2
1
(B) 5_
3
(C) 6
(D) 12
(E) 34
College Entrance Exam Practice
ge07se_c04_0284_0293.indd 289
1
3. Encourage students to draw a rough
sketch of the triangle to estimate the
position of the vertices. This should
eliminate choices A and E. Using the
coordinates, slope, and the Distance
Formula eliminates choices B and C.
289
12/3/05 7:03:09 PM
4. Students may choose answer G because
they found the measures of the remote
interior angles and subtracted their sum
from 180. The result is the measure of
the third interior angle instead of the
exterior angle labeled y.
5. Students may choose answer D because
they did not answer the question being
asked. Remind students to read each
test item carefully.
College Entrance Exam Practice
289
CHAPTER
4
Organizer
Extended Response: Write Extended Responses
Objective: Provide opportunities
Extended-response questions are designed to assess
your ability to apply and explain what you have learned.
These test items are graded using a 4-point
scoring rubric.
GI
to learn and practice common testtaking strategies.
<
D@<I
Online Edition
Resources
State Test Prep CD-ROM
Extended Response Given
p q, state which theorem, AAS,
ASA, SSS, or SAS, you would use
to prove that ABC DCB.
Explain your reasoning.
State Test Practice Online
4-point response:
State Test Prep Workbook
µ
Scoring Rubric
4 points: The student shows an
understanding of properties relating to
parallel lines, triangle congruence, and the
differences between ASA, SSS, and SAS.
3 points: The student correctly chooses
which theorem to use but does not
completely defend the choice or leaves out
crucial understanding of parallel lines.
2 points: The student chooses the correct
theorem but only defends part of it.
«
1 point: The student does not follow
directions or does not provide any
explanation for the answer.
L`][gjj][ll`]gj]elgmk]akK9K&9[[gj\af_lg
0 points: The student does not attempt
l`]^a_mj]$9 ;5 <:&:ql`]J]^d]pan]Hjgh]jlq$
to answer.
:;5 :;&Kgalbmklf]]\klgZ]k`gofl`Yl
:;9 5 ;:<&Kaf[]hddiYf\l`]qYj][mlZq
ljYfkn]jkYd:;$ :;9 5 ;:<Zql`]9dl]jfYl]Afl]jagj9f_d]kL`]gj]e&KgZqK9K$
9:; 5 <;:&KKK[YffglZ]mk]\lghjgn]l`Yl 9:; 5 <;:Z][Ymk]al[Yffgl
Z]hjgn]fl`Yl9: 5 <;&9K9[YffglZ]mk]\Z][Ymk]al[YffglZ]hjgn]fl`Yl
:9; 5 ;<:&99K[YffglZ]mk]\Z][Ymk]al[YffglZ]hjgn]fl`Yl9:; 5 <;:&
KEYWORD: MG7 TestPrep
This Test Tackler
explains how
extended-response
test items are scored and demonstrates how to create a response that
is deserving of full credit. Explain
to students that extended response
questions are longer and more complex than short-response questions.
The scoring rubric used is based on
a 4-point scale rather than a 2-point
scale. Point out how these types of
questions may have different parts.
For full credit, each part of the question has to be completed correctly
and explained thoroughly.
The student gave a complete, correct response to the question and provided
an explanation as to why the other theorems could not be used.
3-point response:
The reasoning is correct, but the student did not explain why other theorems could
not be used.
2-point response:
The answer is correct, but the student did not explain why the included angles
are congruent.
1-point response:
The student did not provide any reasoning.
290
Chapter 4 Triangle Congruence
ge07se_c04_0284_0293.indd 290
290
Chapter 4
12/3/05 7:03:14 PM
To receive full credit, make sure all parts of the
problem are answered. Be sure to provide a
complete explanation for your reasoning.
Read each test item and answer the questions
that follow.
Answers
Item B
Can an equilateral triangle be an obtuse
triangle? Explain your answer. Include a sketch
to support your reasoning.
Possible answers:
1. the name of each thm. used
and why
2. 1 pt.; although the student provided a partial correct response,
no reasoning is provided.
5. What should a full-credit response to this
test item include?
6. A student wrote this response:
Scoring Rubric:
3. all 6 pairs of corr. parts; SSS;
SAS; ASA; AAS; HL; no
4 points: The student demonstrates a thorough
understanding of the concept, correctly
answers the question, and provides a complete
explanation.
4. You can use the Pyth. Thm. to
−− −−
prove NP YZ. Then once you
have shown that corr. sides of
each are , you can use SSS
to prove MNP XYZ.
3 points: The student correctly answers the
question but does not show all work or does not
provide an explanation.
2 points: The student makes minor errors
resulting in an incorrect solution but shows and
explains an understanding of the concept.
5. an explanation of the answer to
the question and a sketch of an
equil. and an obtuse 1 point: The student gives a response showing
no work or explanation.
0 points: The student gives no response.
6. The response is not complete,
and there is no sketch provided.
Why will this response not receive a score
of 4 points?
7. Since an obtuse must include
an obtuse ∠, and an equil. contains only acute , it cannot
possibly be an obtuse .
7. Correct the response so that it receives
full credit.
Item A
What theorem(s) can you use, other than the
HL Theorem, to prove that MNP XYZ ?
Explain your reasoning.
Item C
An isosceles right triangle has two sides, each
with length y + 4.
<
*
8
9
1. What should a full-credit response to this
test item include?
2. A student wrote this response:
Describe how you would find the length of
the hypotenuse. Provide a sketch in your
explanation.
8. No; the question asks for the student to describe how to find the
length of the hyp., not to actually
do the calculation.
8. A student began trying to find the length of
the hypotenuse by writing the following:
9. I would draw a sketch of an
isosc. rt. with side lengths
labeled and then describe, using
the Pyth. Thm., how to find the
length of the hyp.
What score should this response receive?
Why?
3. Write a list of the ways to prove triangles
congruent. Is the Pythagorean Theorem on
your list?
4. Add to the response so that it receives
a score of 4-points.
Is the student on his way to receiving a
4-point response? Explain.
9. Describe a different method the student
could use for this response.
Test Tackler
ge07se_c04_0284_0293.indd 291
291
12/3/05 7:03:21 PM
KEYWORD: MG7 Resources
Test Tackler
291
CHAPTER
4
KEYWORD: MG7 TestPrep
Organizer
CUMULATIVE ASSESSMENT, CHAPTERS 1–4
practice for Chapters 1—4 and
standardized tests.
Multiple Choice
GI
Objective: Provide review and
<D
@<I
6. Which conditional statement has the same truth
value as its inverse?
Use the diagram for Items 1 and 2.
If n < 0, then n 2 > 0.
Online Edition
If a triangle has three congruent sides, then
it is an isosceles triangle.
Resources
If n is a negative integer, then n < 0.
Assessment Resources
1. Which of these congruence statements can be
Chapter 4 Cumulative Test
proved from the information given in the figure?
State Test Prep Workbook
AEB CED
ABD BCA
BAC DAC
DEC DEA
State Test Prep CD-ROM
2. What other information is needed to prove that
State Test Practice Online
CEB AED by the HL Congruence Theorem?
−− −−−
−−− −−
AD AB
CB AD
−− −−
−− −−
BE AE
DE CE
KEYWORD: MG7 TestPrep
3. Which biconditional statement is true?
Tomorrow is Monday if and only if today is
not Saturday.
For Item 7, students
who chose B added
the difference in the
x-values of the points to the difference in the y-values of the points.
Students who chose C divided by
2 when they should have found a
square root.
Next month is January if and only if this
month is December.
Today is a weekend day if and only if
yesterday was Friday.
This month had 31 days if and only if last
month had 30 days.
intersects ST
at more
4. What must be true if PQ
For Item 8, encourage students to
rule out the answer choices that do
not have the correct y-intercept. This
will eliminate choices H and J.
than one point?
For Item 14, suggest that students
draw a diagram. They can use the
diagram to help identify corresponding parts of the congruent triangles.
9.0 miles
6.0 miles
15.8 miles
8. A line has an x-intercept of -8 and a y-intercept
of 3. What is the equation of the line?
8x - 8
y = -8x + 3
y=_
3
3
_
y= x+3
y = 3x - 8
8
passes through points J(1, 3) and K(-3, 11).
9. JK
?
Which of these lines is perpendicular to JK
1
1
1x + _
y = -_
y = -2x - _
5
3
2
1x + 6
y=_
y = 2x - 4
2
10. If PQ = 2(RS) + 4 and RS = TU + 1, which
equation is true by the Substitution Property
of Equality?
PQ = TU + 5
PQ = 2(TU) + 5
and ST
are opposite rays.
PQ
PQ = 2(TU) + 6
5. ABC DEF, EF = x 2 - 7, and BC = 4x - 2.
Find the values of x.
-1 and 5
1 and 5
-1 and 6
2 and 3
11. Which of the following is NOT valid for proving
that triangles are congruent?
AAA
SAS
ASA
HL
Chapter 4 Triangle Congruence
20. Possible answer: Because the acute
of a rt. are comp., ∠1 is comp.
to ∠2. By the Corr. Post., ∠1 ∠3.
Therefore ∠3 is also comp. to ∠2 by
the Comps. Thm.
21a. 2x + 12 + x = 180
3x + 12 = 180 (Simplify.)
3x + 12 - 12 = 180 - 12 (Subtr.
Prop. of =)
3x = 168 (Simplify.)
3x
168
=
(Div. Prop. of =)
3
3
x = 56 (Simplify.)
_ _
Chapter 4
4.2 miles
PQ = TU + 6
ge07se_c04_0284_0293.indd 292
292
a reef has coordinates (6, 8). If each map unit
represents 1 mile, what is the distance between
the island and the reef to the nearest tenth of a
mile?
P, Q, S, and T are noncoplanar.
Answers
KEYWORD: MG7 Resources
7. On a map, an island has coordinates (3, 5), and
P, Q, S, and T are collinear.
are perpendicular.
and ST
PQ
292
If an angle measures less than 90°, then it is
an acute angle.
21b. Possible answer: The sum of the
measures of an ∠ and its comp. is
90°. Therefore any ∠ that measures
90° or greater does not have a comp.
Because m∠H = x° = 56°, the ∠
does have a comp. Because m∠G =
(2x + 12)° = [2(56) + 12]° = 124°,
∠G does not have a comp.
22a. 90; based on the conjecture, 60 out of
1000 parts will be defective. Express
this ratio as a percent:
60
____
1000
=
6
___
= 6%. Find 6% of 1500.
100
1500 × 6% = 1500 × 0.06 = 90.
Based on the conjecture, 90 out of
1500 parts will be defective.
12/3/05 7:03:26 PM
Use this diagram for Items 12 and 13.
Short Response
Short-Response Rubric
20. Given m with transversal n, explain why ∠2
Items 20–23
and ∠3 are complementary.
£ääÂ
2 Points = The student’s answer is
an accurate and complete execution of the task or tasks.
˜
Î
£
12. What is the measure of ∠ACD?
40°
100°
80°
140°
Ű
1 Point = The student’s answer contains attributes of an appropriate
response but is flawed.
Ó
“
0 Points = The student’s answer
contains no attributes of an appropriate response.
21. ∠G and ∠H are supplementary angles.
m∠G = (2x + 12)°, and m∠H = x°.
13. What type of triangle is ABC?
a. Write an equation that can be used to
determine the value of x. Solve the equation
and justify each step.
Isosceles acute
Equilateral acute
Extended-Response
Rubric
b. Explain why ∠H has a complement but ∠G
Isosceles obtuse
does not.
Scalene acute
Item 24
22. A manager conjectures that for every 1000 parts
Take some time to learn the directions for filling
in a grid. Check and recheck to make sure you are
filling in the grid properly. You will only get credit
if the ovals below the boxes are filled in correctly.
To check your answer, solve the problem using
a different method from the one you originally
used. If you made a mistake the first time, you
are unlikely to make the same mistake when
you solve a different way.
Gridded Response
14. CDE JKL. m∠E = (3x + 4)°, and
m∠L = (6x - 5)°. What is the value of x? 3
a factory produces, 60 are defective.
how many of them can be expected to be
defective based on the manager’s conjecture?
Explain how you found your answer.
b. Use the data in the table below to show that
the manager’s conjecture is false.
Day
1
2
3
4
5
Parts
1000
2000
500
1500
2500
60
150
30
90
150
Defective
Parts
the same street. Eduardo’s house is halfway
between Lucy’s house and Frank’s house.
Lucy’s house is halfway between Carmen’s house
and Frank’s house. If the distance between
Eduardo’s house and Lucy’s house is 150 ft,
what is the distance in feet between Carmen’s
house and Eduardo’s house? 450 ft
16. JKL XYZ, and JK = 10 - 2n. XY = 2, and
YZ = n 2. Find KL.
16
3 Point = The student answers correctly, but the explanations may
contain minor flaws. Work demonstrates an understanding of major
concepts related to isosceles and
congruent triangles.
−−
−−
15. Lucy, Eduardo, Carmen, and Frank live on
4 Points = The student correctly
finds m∠D and the value of x and
shows that ABC DEF .
Explanations are complete, and
work demonstrates a thorough
understanding of concepts related
to isosceles and congruent triangles.
a. If the factory produces 1500 parts in one day,
23. BD is the perpendicular bisector of AC.
a. What are the conclusions you can make from
this statement?
−−
−−
2 Points = The student answers
correctly, but the explanations are
missing or incomplete. Or the student answers part of the problem
only. Work demonstrates a limited
understanding of concepts.
−−
b. Suppose BD intersects AC at D. Explain why BD
−−
is the shortest path from B to AC.
Extended Response
−−
−− −−
and AC DF. m∠C = 42.5°, and m∠E = 95°.
−−
24. ABC and DEF are isosceles triangles. BC EF,
1 Point = The student answers
incorrectly but makes a reasonable attempt to show work or offer
explanation.
a. What is m∠D? Explain how you determined
17. An angle is its own supplement. What is
its measure?
90°
18. The area of a circle is 154 square inches.
What is its circumference to the nearest inch?
44 in.
19. The measure of ∠P is 3__12 times the measure of ∠Q.
If ∠P and ∠Q are complementary, what is m∠P
in degrees? 70
your answer.
b. Show that ABC and DEF are congruent.
c. Given that EF = 2x + 7 and AB = 3x + 2, find
0 Points = The student does not
answer correctly and does not
attempt all parts of the problem.
the value for x. Explain how you determined
your answer.
23a. Possible answer: ∠BDA is a rt. ∠;−−∠BDC
is a rt. ∠; AD = DC; D is the mdpt. of AC.
23b. The shortest dist. from a pt. to a line is
measured along the ⊥ from the pt. to the line.
Cumulative Assessment, Chapters 1–4
Answers
22b. Possible answer: Based on the manager’s conjecture, 120 parts should be
defective if 2000 parts are produced.
However, the table shows that 150
out of 2000 parts were defective on
day 2. Therefore the data for day 2 is
a counterexample to the manager’s
conjecture. This shows that the conjecture is false.
ge07se_c04_0284_0293.indd 293
24a. 42.5°
Possible answer: Because DEF is an
isosc. , 2 of its sides are , and the
opp. these sides are . ∠E is an
obtuse ∠. Because a cannot
have more than 1 obtuse ∠, the 2 in DEF must be ∠D and ∠F. Use
this information to find m∠D. m∠D +
m∠E + m∠F = 180° by the Sum
Thm. x + 95° + x = 180°. Substitute x
for m∠D and m∠F, and 95° for m∠E.
2x = 85°. Simplify and then subtract
95° from both sides. x = 42.5°. Divide
both sides by 2. Thus m∠D = 42.5°.
293
24c. 5; possible answer: because ∠D −− −−
∠F, DE EF by the Conv. of the Isosc.
Thm.
−− −−
5/8/06
PM
AB 1:09:39
DE
CPCTC
−− −−
AB EF
Trans. Prop. of AB = EF
Def. of Segs.
3x + 2 = 2x + 7
Subst. Prop. of =
x=5
Subtr. Prop. of =
24b. Possible answer: From part a, ∠F
∠D. Therefore m∠F = m∠D =
42.5°. It is given that m∠C = 42.5°.
Therefore ∠C ∠F. It is also given
−− −−
−− −−
that BC EF and AC DF. Therefore
ABC DEF by SAS.
Cumulative Assessment
293
Problem Solving
on Location
Organizer
MICHIGAN
GI
Objective: Choose appropriate
problem-solving strategies
and use them with skills from
Chapters 3 and 4 to solve
real-world problems.
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Online Edition
The Queen’s Cup
The annual Queen’s Cup race is one of the most exciting sailing events of the
year. Traditionally held at the end of June, the race attracts hundreds of yachts
that compete to cross Lake Michigan—at night—in the fastest time possible.
The Queen’s Cup
Reading
Strategies
+ALAMAZOO
Choose one or more strategies to solve each problem.
ENGLISH
LANGUAGE
LEARNERS
1. The race starts in Milwaukee, Wisconsin,
and ends in Grand Haven, Michigan.
The boats don’t sail from the start to the
finish in a straight line. They follow a zigzag
course to take advantage of the wind.
Suppose one of the boats leaves Milwaukee
at a bearing of N 50° E and follows the
course shown. At what bearing does the
boat approach Grand Haven? S 20° E
Have students read Problem 2. Then
suggest that they reread the problem, this time jotting down the given
information as they read.
Using Data The table on this page
uses bearings. Have one or more
students explain how bearings are
measured. As they do so, ask them
to sketch some examples on the
board.
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2. The Queen’s Cup race is 78.75 miles long. In 2004, the winning
sailboat completed the first 29.4 miles in about 3 hours and
the first 49 miles in about 5 hours. Suppose it had continued
at this rate. What would the winning time have been? 8 h
3. During the race one of the boats leaves Milwaukee M,
sails to X, and then sails to Y. The team discovers a problem
with the boat so it has to return directly to Milwaukee.
Does the table contain enough information to determine
the course to return to M? Explain.
Bearing
Distance (mi)
M to X
N 42° E
3.1
X to Y
S 59° E
2.4
Y to M
294
Ask students what strategy they would use
to solve Problem 2. The strategy Make a
Table may be especially useful in organizing the given information.
ge07se_c04_0294_0295.indd 294
294
Chapter 4
9
Chapter 4 Triangle Congruence
Problem-Solving Focus
KEYWORD: MG7 Resources
8
Discuss with students different ways they
might find the winning time. Some students may find it helpful to use proportional reasoning. To foster this type of thinking,
encourage students to rephrase the problem in if-then form. If it takes 3 hours to go
29.4 miles and 5 hours to 49 miles, then
it takes x hours to go 29.4 miles. Other
students may benefit from thinking about
this problem in terms of slope. Ask them
what ordered pairs they could use to find
the slope of the line that represents the
boat’s distance at a given time. (3,29.4)
and (5,49)
Answers
3. Yes; there is enough information to find
m∠MXY (101°). MX and MY are known,
so a unique MXY is determined by
SAS.
9/5/05 1:47:36 PM
Problem
Solving
Strategies
Draw a Diagram
Make a Model
Guess and Test
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
The Air Zoo
Located in Kalamazoo,
Michigan, the Air Zoo offers
visitors a thrilling, interactive
voyage through the history of flight.
It features full-motion flight
simulators, a “4-D” theater,
and more than 80 rare aircraft.
The Air Zoo is also home to
The Century of Flight, the
world’s largest indoor mural.
Choose one or more strategies to
solve each problem.
Months of Work
Amount
Completed (ft 2)
2
5,236
3. The Air Zoo’s flight simulators let
visitors practice takeoffs and landings.
To determine the position of a plane
during takeoff, an airport uses two
cameras mounted 1000 ft apart.
What is the distance d that the plane
has moved along the runway since
it passed camera 1?
850 ft
5
13,091
7
18,327
Make sure that students know what
a mural is. Spanish-speaking ELL
students may recognize the similarity of the word mural to muro, the
Spanish word for wall.
Encourage students to use the fourstep problem-solving process for the
problems. Focus on the third step:
(3) Plan. In particular, ask students
to consider the information that is
given in the statement of Problem 2,
and in the accompanying diagram.
2. Visitors to the Air Zoo can see a
replica of a Curtiss JN-4 “Jenny,”
the plane that flew the first official
U.S. airmail route in 1918. The plane
−−
−−
has two parallel wings AB and CD
that are connected by bracing wires.
The wires are arranged so that
−−
m∠EFG = 29° and GF bisects ∠EGD.
What is m∠AEG? 58°
Discuss Ask students to decide
which pieces of information are
−−
essential in solving the problem. AB
−−
−−
CD; m∠EFG = 29°; GF bisects
∠EGD Then have students describe
a sequence of steps they can use to
find m∠AEG. First find the m∠FGD.
Then add m∠FGD and m∠EGF.
Finally use the Alt. Int. ∠ Thm.
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Problem Solving on Location
ge07se_c04_0294_0295.indd 295
ENGLISH
LANGUAGE
LEARNERS
Problem-Solving Focus
әÂ
Reading
Strategies
Using Data Before students begin
Problem 1, have a brief discussion
about the data in the table. Ask
students to estimate the number
of months it took to complete the
mural. approximately 11 Later they
can compare their calculated values
to these initial estimates.
Painting The Century of Flight
1. The Century of Flight mural
measures 28,800 square feet—
approximately the size of three
football fields! The table gives data
on the rate at which the mural was
painted. How many months did it
take to complete the mural? ≈ 11 mo
The Air Zoo
295
12/20/05 3:24:19 PM
Problem Solving on Location
295